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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_003.html#V11" title="GLIB_003:attr.11">real-WEV</a> <a href="glib_003.html#NM7" title="GLIB_003:NM.7">WEVGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/>

<b>set </b><font color="Maroon" title="c2">NextG</font> =  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">BestEdges</font> =  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">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> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">NextEG</font> = <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1;<br/>
<b>set </b><font color="Maroon" title="c6">target</font> = <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_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">val</font> = <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> <font color="Maroon" title="c1">G</font></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> <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_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
(  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/>
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <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" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_1"/>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</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#D5" target="_self" title="GLIB_004:def.5">Def5</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="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/><div class="add"><a NAME="E2:29_1_1_2"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
 <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<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="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def5</a></i>;<br/><a NAME="E3:29_1_1_2"/><i><font color="Green" title="E20">A3</font></i>: 
 <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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:29_1_1_2"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/><a NAME="E4:29_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41" title="GLIB_003:th.41">GLIB_003:41</a></i>;<br/><a NAME="E5:29_1_1_2"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a>, <a class="ref" href="glib_003.html#T43" title="GLIB_003:th.43">GLIB_003:43</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">target</font> = <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_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="glib_004.html#E3:29_1_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E7:29_1_1_2"/>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1 &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:29_1_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="glib_003.html#T41" title="GLIB_003:th.41">GLIB_003:41</a>, <a class="ref" href="glib_003.html#T47" title="GLIB_003:th.47">GLIB_003:47</a></i>;<br/><b>hence </b><a NAME="E8:29_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T88" title="GLIB_000:th.88">GLIB_000:88</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/><b>thus </b> <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> = 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#T42" title="GLIB_003:th.42">GLIB_003:42</a></i>
<br/>.= 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E2:29_1_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="glib_003.html#T48" title="GLIB_003:th.48">GLIB_003:48</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  )</span><br/><b>assume </b><a NAME="E1:29_1_1_2_2"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span><br/><a NAME="E2:29_1_1_2_2"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:29_1_1_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="glib_003.html#T53" title="GLIB_003:th.53">GLIB_003:53</a></i>;<br/><a NAME="E3:29_1_1_2_2"/>
 <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:29_1_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="glib_003.html#T49" title="GLIB_003:th.49">GLIB_003:49</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:29_1_1_2_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="glib_004.html#E2:29_1_1_2_2"><i><font color="Green" title="E23">A6</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><b>hence </b><a NAME="E11:29_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )</span><br/><b>assume </b><a NAME="E1:29_1_1_2_3"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span><br/><a NAME="E2:29_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</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"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</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_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <font color="Maroon" title="c8">target</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p0">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" title="GLIB_004:func.2">DIJK:NextBestEdges</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T60" title="GLIB_003:th.60">GLIB_003:60</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_2_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</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#E2:29_1_1_2"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="glib_004.html#E5:29_1_1_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="glib_004.html#E1:29_1_1_2_3"><i><font color="Green" title="E22">A7</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><b>hence </b><a NAME="E13:29_1_1_2"/>
<font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></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#D3" title="TARSKI:def.3">TARSKI:def 3</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="E2:29"/>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <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"><a href="glib_004.html#K3" title="GLIB_004:func.3">DIJK:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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