<?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>;<br/><b>let </b><font color="Maroon" title="c2">src</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">DCS</font> =  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>;<br/>
<b>set </b><font color="Maroon" title="c5">RFS</font> = <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c7">Gn1a</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">VGn</font> =  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">BestEdges</font> =  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">SG</font> =  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c12">SGn</font> =  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">TG</font> =  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c14">TGn</font> =  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#D7" target="_self" title="GLIB_004:def.7">Def7</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">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> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/><a NAME="E2:35_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T18" target="_self" title="GLIB_004:th.18">Th18</a></i>;<br/><a NAME="E3:35_1_1"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:35_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  );<br/><b>consider </b><font color="Maroon" title="c15">BE1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:35_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">BE1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1" title="SUBSET_1:sch.1">SUBSET_1:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c15">BE1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c16">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green" title="E29">A7</font></i>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> &amp; not <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:35_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">v</font> = <font color="Maroon" title="c16">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="glib_004.html#E3:35_1_1_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">W</font> being  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:35_1_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c18">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c17">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:35_1_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="glib_002.html#D6" title="GLIB_002:def.6">GLIB_002:def 6</a></i>;<br/><a NAME="E7:35_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K4" title="GLIB_004:func.4">DIJK:Init</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#D7" target="_self" title="GLIB_004:def.7">Def7</a></i>;<br/><a NAME="E8:35_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <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="c2">src</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_004.html#T15" target="_self" title="GLIB_004:th.15">Th15</a></i>;<br/><a NAME="E9:35_1_1_1"/><b>then </b><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c2">src</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:35_1_1_1"/><i><font color="Green" title="E32">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T16" target="_self" title="GLIB_004:th.16">Th16</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( not $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">W</font> &amp; not <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  );<br/><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">W</font></span>)</span> = 
<font color="Maroon" title="c18">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<b>by </b><i><a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c17">v</font>
<b>by </b><i><a class="txt" href="glib_004.html#E6:35_1_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/><a NAME="E12:35_1_1_1"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E3:35_1_1_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:35_1_1_1"/><i><font color="Green" title="E34">A12</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">k</font>] &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="glib_004.html#E12:35_1_1_1"><i><font color="Green" title="E33">A11</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_2_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_2_1_1"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> = 
<font color="Maroon" title="c18">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c2">src</font>
<b>by </b><i><a class="txt" href="glib_004.html#E6:35_1_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/><b>hence </b><a NAME="E3:35_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E9:35_1_1_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="glib_004.html#E10:35_1_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_2_1_2"/><i><font color="Green" title="E35">A13</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c20">k'</font> = <font color="Maroon" title="c19">k</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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>by </b><i><a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:35_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E4:35_1_1_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:35_1_1_1_2_1_2"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:35_1_1_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:35_1_1_1_2_1_2"/><b>then </b>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">k2a</font> = <font color="Maroon" title="c20">k'</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E8:35_1_1_1_2_1_2"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">W</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E9:35_1_1_1_2_1_2"/>
<font color="Maroon" title="c21">k2a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E10:35_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k2a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="glib_004.html#E8:35_1_1_1_2_1_2"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c22">e</font> = <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E11:35_1_1_1_2_1_2"/>
<font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k2a</font>,<font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:35_1_1_1_2_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="glib_001.html#T123" title="GLIB_001:th.123">GLIB_001:123</a></i>;<br/><a NAME="E12:35_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E38">A16</font></i>: 
( <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &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"><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2a</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> <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k2a</font> &amp; <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"><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2a</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> <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E13:35_1_1_1_2_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T17" target="_self" title="GLIB_004:th.17">Th17</a></i>;<br/><a NAME="E14:35_1_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2a</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> <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k2a</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">k2a</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> <font color="Maroon" title="c18">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E12:35_1_1_1_2_1_2"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E15:35_1_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E5:35_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="glib_004.html#E1:35_1_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="glib_004.html#E14:35_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="glib_004.html#E10:35_1_1_1_2_1_2"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:35_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c16">BE1</font> = <font color="Maroon" title="c15">BE1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">BE1</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c17">e1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">BE1</font><b> such that </b><br/><a NAME="E9:35_1_1"/><i><font color="Green" title="E28">A17</font></i>: 
for <font color="Olive" title="b1">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">BE1</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">e1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e2</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></i><a NAME="E10:35_1_1"/><i><font color="Green" title="E29">A18</font></i>: 
( <font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E5:35_1_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/><a NAME="E11:35_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E12:35_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:35_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:35_1_1"/><b>then </b><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c17">e1</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:35_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="glib_000.html#D18" title="GLIB_000:def.18">GLIB_000:def 18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_2"/>
<font color="Maroon" title="c18">y</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E2:35_1_1_2"/><b>then </b><i><font color="Green" title="E31">A20</font></i>: 
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D18" title="GLIB_000:def.18">GLIB_000:def 18</a></i>;<br/><a NAME="E3:35_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:35_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">BE1</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:35_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="glib_004.html#E2:35_1_1_2"><i><font color="Green" title="E31">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E9:35_1_1"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:35_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E1:35_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="glib_004.html#E13:35_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="glib_004.html#D4" target="_self" title="GLIB_004:def.4">Def4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:35_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:35"/><i><font color="Green" title="E24">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/>
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">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> 
 ;<br/><a NAME="E2:35_2"/><b>then </b><i><font color="Green" title="E25">A22</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </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"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</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="glib_004.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="glib_004.html#E3:35"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="glib_004.html#T13" target="_self" title="GLIB_004:th.13">Th13</a></i>;<br/><a NAME="E3:35_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T18" target="_self" title="GLIB_004:th.18">Th18</a></i>;<br/><a NAME="E4:35_2"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/><a NAME="E5:35_2"/><b>then </b>
<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"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</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="txt" href="glib_004.html#E2:35_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/><b>hence </b><a NAME="E6:35_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">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> 
 ;<br/>


</div>
