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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
(  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/>
 <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></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">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:27_1_1_2"/><i><font color="Green">E20</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:27_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E4:27_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41">GLIB_003:41</a></i>;<br/><a NAME="E5:27_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> &amp;  <a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_003.html#T43">GLIB_003:43</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <b>by </b><i><a class="txt" href="glib_004.html#E3:27_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E7:27_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1 <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:27_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="glib_003.html#T41">GLIB_003:41</a>, <a class="ref" href="glib_003.html#T47">GLIB_003:47</a></i>;<br/><b>hence </b><a NAME="E8:27_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>thus </b> <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#T42">GLIB_003:42</a></i>
<br/>.= 
 <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E2:27_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="glib_003.html#T48">GLIB_003:48</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><a NAME="E2:27_1_1_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:27_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_003.html#T53">GLIB_003:53</a></i>;<br/><a NAME="E3:27_1_1_2_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:27_1_1_2_2"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E4:27_1_1_2_2"/><i><font color="Green">E24</font></i>: 
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:27_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="glib_003.html#T49">GLIB_003:49</a></i>;<br/><a NAME="E5:27_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:27_1_1_2_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E6:27_1_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E4:27_1_1_2_2"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:27_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:27_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E3:27_1_1_2_3"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E5:27_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E4:27_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">c<sub>10</sub></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">the_VLabel_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p0">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T60">GLIB_003:60</a></i>;<br/><b>hence </b><a NAME="E5:27_1_1_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:27_1_1_2_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="glib_004.html#E2:27_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:27_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:27"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/>


</div>
