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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EL</font> being  <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">source</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Olive" title="b1">EL</font>,<font color="Olive" title="b2">source</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a></span><br/><b>let </b><font color="Maroon" title="c2">EL</font> be   <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">source</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Olive" title="b1">source</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a></span><br/><b>let </b><font color="Maroon" title="c3">source</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a></span><br/>





<b>set </b><font color="Maroon" title="c4">CS</font> =  <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E23">A1</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 <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E2:28_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th3</a></i>;<br/><a NAME="E3:28_1"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>;<br/><div><i><font color="Green" title="E25">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>set </b><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font>;<br/><b>set </b><font color="Maroon" title="c7">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E2:28_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="glib_005.html#K3" title="GLIB_005:func.3">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D12" target="_self" title="GLIB_005:def.12">Def15</a></i>;<br/><b>set </b><font color="Maroon" title="c8">Next</font> =  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1_1"/>
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <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="c1">G</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> ) or (  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <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="c1">G</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_1"/>
 <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:28_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E23">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_2"/><i><font color="Green" title="E28">A6</font></i>: 
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <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="c1">G</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:28_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><a NAME="E3:28_1_1_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1_1_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="glib_005.html#E1"><i><font color="Green" title="E1">Tw0</font></i></a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:28_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E1:28_1_1_1_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <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="c1">G</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:28_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><a NAME="E3:28_1_1_1_1_3"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="txt" href="#E1:28_1_1_1_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def13</a></i>;<br/><a NAME="E4:28_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:28_1_1_1_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def10</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def11</a></i>;<br/><a NAME="E5:28_1_1_1_1_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_1_1_1_1_3"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="glib_005.html#E1"><i><font color="Green" title="E1">Tw0</font></i></a></i>;<br/><b>hence </b><a NAME="E6:28_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:28_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E4:28_1_1_1_1_3"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:28_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:28_1"/><i><font color="Green" title="E26">A12</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="#E3:28_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E4:28_1"><i><font color="Green" title="E25">A3</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c5">x</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<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>;<br/><a NAME="E6:28_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</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></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E5:28_1"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/><a NAME="E7:28_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<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="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/><b>hence </b><a NAME="E8:28_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28"/>
 <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D56" title="GLIB_000:def.56">GLIB_000:def 56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
