<?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#V9">real-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>;<br/><b>let </b><font color="Maroon">source</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><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>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>set </b><font color="Maroon">CS</font> =  <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>set </b><font color="Maroon">FAP</font> =  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> ;<br/>
<b>set </b><font color="Maroon">GL</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">GL1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:47"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<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>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>;<br/>
<a NAME="E2:47"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:47_1"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">Next</font> =  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_1"/>
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; 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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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="47_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1_1"/>
 <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:47_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:47_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</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="47_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_1_2"/><i><font color="Green">E92</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; 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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:47_1_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_1_1_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:47_1_1_1_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:47_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E5:47_1_1_1_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:47_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_003.html#T50">GLIB_003:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:47_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_3"><b>suppose </b></a><a NAME="E1:47_1_1_1_3"/><i><font color="Green">E96</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:47_1_1_1_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:47_1_1_1_3"/><i><font color="Green">E110</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1_1_1_3_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:47_1_1_1_3_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_1_1_1_3_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:47_1_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E5:47_1_1_1_3_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E6:47_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_003.html#T50">GLIB_003:50</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:47_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</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="E4:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:47"/><b>then </b><i><font color="Green">E128</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:47"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:47"/>
 <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font> is <a href="glib_000.html#V13">halting</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:47"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>;<br/>
<a NAME="E8:47"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E9:47"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E132</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:47_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:47_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E11:47"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_3_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  or <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_1"><b>suppose </b></a><a NAME="E1:47_3_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:47_3_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:47_3_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:47_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_1_2"><b>suppose </b></a><a NAME="E1:47_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:47_3_1_2"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:47_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:47_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:47"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/>


</div>
