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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <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#K2" target="_self">DIJK:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</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>4</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>5</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>6</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>7</sub></font> =  <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>;<br/>
<b>set </b><font color="Maroon">c<sub>8</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>9</sub></font> =  <a href="card_1.html#K4">card</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/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_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>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<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="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>13</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>14</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><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E17</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1"/><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/><a NAME="E2:26_1_1"/><b>then </b><i><font color="Green">E19</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>
 ;<br/><a NAME="E3:26_1_1"/>
<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="E4:26_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</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#E2:26_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:26_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <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#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T63">GLIB_003:63</a></i>;<br/><a NAME="E7:26_1_1"/>
 <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="glib_000.html#R4">DSJoins</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:26_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="glib_004.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E8:26_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E9:26_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>15</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#E6:26_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:26_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</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>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><span class="p0">(<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="p0">(<span class="default"><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></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<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="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></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</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>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E9:26_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="glib_003.html#T61">GLIB_003:61</a></i>;<br/><a NAME="E11:26_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</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>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E1:26_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="glib_004.html#E1:26_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="glib_004.html#E10:26_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:26_1_1"/>
 <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></i>;<br/><a NAME="E13:26_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</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> <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>
 ;<br/><a NAME="E14:26_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E11:26_1_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E15:26_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:26_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>
<b>end;</b></div>

<b>thus </b><a NAME="E2:26"/>
(  <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>  implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E17</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2_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/><a NAME="E2:26_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E1:26_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E3:26_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:26_2"/>
 <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>
<b>end;</b></div>

<b>assume </b><a NAME="E4:26"/><i><font color="Green">E17</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/>

<a NAME="E5:26"/><i><font color="Green">E18</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#E4:26"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E6:26"/>
<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="E7:26"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</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#E5:26"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E9:26"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K7">the_VLabel_of</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>15</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>
 
<b>by </b><i><a class="txt" href="glib_004.html#E4:26"><i><font color="Green">E17</font></i></a>, <a class="ref" href="glib_004.html#D5" target="_self">Def5</a></i>;<br/>
<b>then </b><i><font color="Green">E19</font></i>: <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>15</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>  = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K3" target="_self">DIJK:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<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/>
<a NAME="E11:26"/>
 <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></i>;<br/>
<b>then </b><i><font color="Green">E20</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</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> <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>

<br/>.= 
<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> 

;<br/>
<a NAME="E13:26"/>
 <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="glib_000.html#R4">DSJoins</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E4:26"><i><font color="Green">E17</font></i></a>, <a class="ref" href="glib_004.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E14:26"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/>
<a NAME="E15:26"/><b>then </b>
not <font color="Maroon">c<sub>15</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#E12:26"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:26"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<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> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E10:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="glib_004.html#E12:26"><i><font color="Green">E20</font></i></a>, <a class="ref" href="glib_003.html#T61">GLIB_003:61</a></i>;<br/>


</div>
