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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V8">nonnegative-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">src</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">DCS</font> =  <a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>;<br/>
<b>set </b><font color="Maroon">D0</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">G2</font> being  <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  holds <br/> ( <font color="Olive">G2</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font> &amp; ( for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/><font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">v</font> ) );<br/>
<b>reconsider </b><font color="Maroon">EL</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/>
<i><font color="Green">E34</font></i>: <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0 = 
 <a href="glib_004.html#K4" target="_self">DIJK:Init</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<span class="p1">(<span class="default"><font color="Maroon">src</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span>

;<br/>
<a NAME="E3:38"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">src</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<a NAME="E4:38"/><b>then </b>
 <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"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">src</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:38"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">src</font></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K3">ELabelSelector</a> 
<b>by </b><i>, <a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D0W</font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E1:38_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E2:38_3"/><b>then </b><i><font color="Green">E49</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">D0W</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">src</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">D0W</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:38_3"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">D0W</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">D0W</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E4:38_3"/><b>then </b>
( <font color="Maroon">D0W</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">D0W</font> <a href="glib_000.html#K18">.size()</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E5:38_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">D0W</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D0W</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:38_3"/>
<font color="Maroon">D0W</font> is <a href="glib_000.html#V4">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D21">GLIB_000:def 21</a></i>;<br/><a NAME="E7:38_3"/><b>then </b>
<font color="Maroon">D0W</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T7">GLIB_002:7</a></i>;<br/><a NAME="E8:38_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">D0W</font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i>, <a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">D0W</font>;<br/><a NAME="E1:38_3_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">D0W</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">x</font>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">D0W</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:38_3_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>let </b><font color="Maroon">W1</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E3:38_3_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E4:38_3_1"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T36">GLIB_003:36</a></i>;<br/><b>thus </b><a NAME="E5:38_3_1"/>
<font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="glib_003.html#T28">GLIB_003:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:38_3"/>
<font color="Maroon">D0W</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E11:38_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E12:38_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">src</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E13:38_3"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">src</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">W1</font> = <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">v</font>;<br/><a NAME="E1:38_3_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">v</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>then consider </b><font color="Maroon">W</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:38_3_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">W</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:38_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">v</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="glib_003.html#T28">GLIB_003:28</a></i>;<br/><a NAME="E5:38_3_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:38_3_2"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="glib_003.html#T36">GLIB_003:36</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:38_3"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:38"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">Dn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Dn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">BE</font> =  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">e</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">source</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">target</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">DnE</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1;<br/><b>set </b><font color="Maroon">pc</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">ec</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">DnW</font> being   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><b>assume </b><a NAME="E2:38_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>let </b><font color="Maroon">Dn1W</font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E3:38_4"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K3" target="_self">DIJK:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:38_4"/><b>then </b><i><font color="Green">E65</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></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"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i/>;<br/><a NAME="E5:38_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E6:38_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E7:38_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">src</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:38_4"/><i><font color="Green">E69</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E9:38_4"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> 0</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"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E10:38_4"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T2" target="_self">Th2</a>, <a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E11:38_4"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:38_4"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E13:38_4"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">DnW</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E14:38_4"/><b>then </b><i><font color="Green">E75</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"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DnW</font> <a href="glib_000.html#K16">.order()</a>  &amp;  <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"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DnW</font> <a href="glib_000.html#K18">.size()</a>  )
 ;<br/><a NAME="E15:38_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">DnW</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font>
 <b>by </b><i/>;<br/><a NAME="E16:38_4"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">DnW</font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i/>;<br/><a NAME="E17:38_4"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">DnW</font> is <a href="glib_002.html#V1">connected</a> &amp; <font color="Maroon">DnW</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">DnW</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_1"/>
(  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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="38_4_1_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_1_1"/>
 <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:38_4_1_1_1"/>
( <font color="Maroon">Dn1W</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font> &amp; ( for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/><font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">v</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2"/><b>then </b><i><font color="Green">E82</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K2" target="_self">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:38_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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="E4:38_4_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E5:38_4_1_1_2"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:38_4_1_1_2"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_003.html#T49">GLIB_003:49</a></i>;<br/><a NAME="E7:38_4_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><a NAME="E8:38_4_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</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"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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="E9:38_4_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_4_1_1_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_1_2_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_4_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:38_4_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E4:38_4_1_1_2_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:38_4_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div><a NAME="E11:38_4_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E12:38_4_1_1_2"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:38_4_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:38"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/><a NAME="E14:38_4_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Dn1W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E15:38_4_1_1_2"/><i><font color="Green">E93</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:38"><i><font color="Green">E47</font></i></a>, </i>;<br/><a NAME="E16:38_4_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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="E17:38_4_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E18:38_4_1_1_2"/><b>then </b>
 <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"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></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="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><a NAME="E19:38_4_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">Dn1W</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">DnW</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E20:38_4_1_1_2"/><i><font color="Green">E97</font></i>: 
not  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/><a NAME="E21:38_4_1_1_2"/><b>then </b>
 <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"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><a NAME="E22:38_4_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">Dn1W</font> <a href="glib_000.html#K18">.size()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">DnW</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E23:38_4_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Dn1W</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Dn1W</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><a NAME="E24:38_4_1_1_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">DnW</font> is    <a href="glib_000.html#M2">Subgraph</a> of <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">Dn1W</font> is    <a href="glib_000.html#M2">Subgraph</a> of <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="glib_004.html#E3:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="glib_000.html#T94">GLIB_000:94</a></i>;<br/><div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:38_4_1_1_2_2"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">DnW</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Dn1W</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:38_4_1_1_2_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font>
 ;<br/><a NAME="E3:38_4_1_1_2_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:38_4_1_1_2_2"/><i><font color="Green">E104</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">DnW</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">DnW</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:38_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="glib_004.html#E5:38_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E5:38_4_1_1_2_2"/>
(  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E6:38_4_1_1_2_2"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">DnW</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">DnW</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:38_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="glib_004.html#E6:38_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="glib_004.html#E7:38_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/></div><b>end;</b></div><a NAME="E26:38_4_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">DnW</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><b>reconsider </b><font color="Maroon">DnW'</font> = <font color="Maroon">DnW</font> as  <a href="glib_003.html#V1">[Weighted]</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E28:38_4_1_1_2"/><i><font color="Green">E106</font></i>: 
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">DnW'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><a NAME="E29:38_4_1_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><a NAME="E30:38_4_1_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E31:38_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="glib_004.html#E8:38_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E32:38_4_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">DnW'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_4_1_1_2_3"/>
<font color="Maroon">y</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#K5">the_Weight_of</a> <font color="Maroon">DnW'</font></span>)</span>
 ;<br/><a NAME="E2:38_4_1_1_2_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">DnW'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="glib_004.html#E6:38_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="glib_004.html#E7:38_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E10:38_4"><i><font color="Green">E71</font></i></a>, <a class="txt" href="glib_004.html#E4:38_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E34:38_4_1_1_2"/><b>then </b>
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">DnW'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E9:38_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E35:38_4_1_1_2"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">DnW</font> is   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><a NAME="E36:38_4_1_1_2"/><i><font color="Green">E219</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E37:38_4_1_1_2"/>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E38:38_4_1_1_2"/><b>then </b><i><font color="Green">E220</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E39:38_4_1_1_2"/>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E40:38_4_1_1_2"/><b>then </b>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E41:38_4_1_1_2"/><b>then </b><i><font color="Green">E221</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:38_4"><i><font color="Green">E73</font></i></a>, <a class="txt" href="glib_004.html#E3:38_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E42:38_4_1_1_2"/><b>then </b><i><font color="Green">E222</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Dn1W</font>, <font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Dn1W</font>;<br/><div><i><font color="Green">E224</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_4_1_1_2_4_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font>, <font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">DnW</font> <b>by </b><i>, <a class="ref" href="glib_004.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon">W1</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">DnW</font><b> such that </b><br/><a NAME="E4:38_4_1_1_2_4_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u'</font>,<font color="Maroon">v'</font>
 <b>by </b><i>, <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W1</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="txt" href="glib_004.html#E5:38_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E6:38_4_1_1_2_4_1"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E7:38_4_1_1_2_4_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_4_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_1_2_4_2_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_4_2_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_4_2_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_4_1_1_2_4_2_1_1"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E15:38_4"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_4_2_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_4_2_1_2"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font><b> such that </b><br/><a NAME="E3:38_4_1_1_2_4_2_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E11:38_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_004.html#E15:38_4"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E4:38_4_1_1_2_4_2_1_2"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="glib_004.html#E14:38_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E5:38_4_1_1_2_4_2_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:38_4_1_1_2_4_2_1_2"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E17:38_4"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_4_2_1_3"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_4_2_1_3"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font><b> such that </b><br/><a NAME="E3:38_4_1_1_2_4_2_1_3"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E11:38_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_004.html#E15:38_4"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E4:38_4_1_1_2_4_2_1_3"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="glib_004.html#E14:38_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E5:38_4_1_1_2_4_2_1_3"/>
<font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v</font>,<font color="Maroon">u</font>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E17:38_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:38_4_1_1_2_4_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K6">.reverse()</a>  <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T24">GLIB_001:24</a></i>;<br/><b>hence </b><a NAME="E7:38_4_1_1_2_4_2_1_3"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_4_2_1_4"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_4_2_1_4"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2_4_2_1_4"/><b>then </b><i><font color="Green">E234</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">Dn1W</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">u</font>;<br/><b>thus </b><a NAME="E3:38_4_1_1_2_4_2_1_4"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:38_4_1_1_2_4"/>
 ex <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">Dn1W</font> st <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div><a NAME="E44:38_4_1_1_2"/><b>then </b>
<font color="Maroon">Dn1W</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><a NAME="E45:38_4_1_1_2"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">Dn1W</font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_5"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">WT</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">DnW</font><b> such that </b><br/><a NAME="E2:38_4_1_1_2_5"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">WT</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">W1</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">WT</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W1</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 <b>by </b><i>, <a class="txt" href="glib_004.html#E11:38_4"><i><font color="Green">E72</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">WT'</font> = <font color="Maroon">WT</font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="txt" href="glib_004.html#E5:38_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">WT'</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:38_4_1_1_2_5"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">WT'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">WT'</font> <a href="glib_001.html#K10">.addEdge</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> as   <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="txt" href="glib_004.html#E13:38_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="glib_001.html#T124">GLIB_001:124</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_5_1"><b>now </b></a><div class="add"><a NAME="E1:38_4_1_1_2_5_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:38_4_1_1_2_5_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Dn1W</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E3:38_4_1_1_2_5_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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">WT'</font> <a href="glib_001.html#K4">.last()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">Dn1W</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:38_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E4:38_4_1_1_2_5_1"/><i><font color="Green">E240</font></i>: 
not  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">DnW</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E9:38"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="glib_004.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E5:38_4_1_1_2_5_1"/>
<font color="Maroon">WT</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">WT'</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/><b>hence </b><a NAME="E6:38_4_1_1_2_5_1"/>
not  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WT'</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E17:38_4"><i><font color="Green">E79</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_4_1_1_2_5_1_1"/><i><font color="Green">E241</font></i>: 
not <font color="Maroon">WT'</font> is <a href="glib_001.html#V3">trivial</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_4_1_1_2_5_1_1_1"/>
<font color="Maroon">WT'</font> is <a href="glib_001.html#V1">closed</a>
 ;<br/><a NAME="E2:38_4_1_1_2_5_1_1_1"/><b>then </b><i><font color="Green">E242</font></i>: 
<font color="Maroon">WT'</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:38_4_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E3:38_4_1_1_2_5_1_1_1"/>
<font color="Maroon">Dn1W</font> is <a href="glib_002.html#V2">acyclic</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E15:38_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="glib_002.html#D3">GLIB_002:def 3</a></i>;<br/><b>hence </b><a NAME="E4:38_4_1_1_2_5_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#D2">GLIB_002:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_4_1_1_2_5_1_1"/>
not <font color="Maroon">WT'</font> is <a href="glib_001.html#V1">closed</a>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:38_4_1_1_2_5_1"/>
( <font color="Maroon">WT'</font> is <a href="glib_001.html#V3">trivial</a> or  not <font color="Maroon">WT'</font> is <a href="glib_001.html#V1">closed</a> )
 ;<br/><a NAME="E9:38_4_1_1_2_5_1"/>
<font color="Maroon">WT'</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">WT</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T99">GLIB_001:99</a></i>;<br/><a NAME="E10:38_4_1_1_2_5_1"/><b>then </b><i><font color="Green">E243</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WT'</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, </i>;<br/><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E11:38_4_1_1_2_5_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">WT'</font> )
 ;<br/><b>hence </b><a NAME="E12:38_4_1_1_2_5_1"/>
<font color="Maroon">WT'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:38_4_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E8:38_4_1_1_2_5"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:38_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><b>reconsider </b><font color="Maroon">W2G</font> = <font color="Maroon">W2</font> as   <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_5_2"><b>now </b></a><div class="add"><a NAME="E1:38_4_1_1_2_5_2"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">WT'</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E2:38_4_1_1_2_5_2"/><i><font color="Green">E246</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Dn1W</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E13:38_4"><i><font color="Green">E74</font></i></a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><b>hence </b><a NAME="E3:38_4_1_1_2_5_2"/>
 <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">WT'</font> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E17:38_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="glib_000.html#T64">GLIB_000:64</a></i>;<br/><a NAME="E4:38_4_1_1_2_5_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="seq_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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Dn1W</font></span>)</span></span>)</span> <a href="seq_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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><b>hence </b><a NAME="E5:38_4_1_1_2_5_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Dn1W</font></span>)</span> <a href="seq_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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="glib_004.html#E1:38_4_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E6:38_4_1_1_2_5_2"/><i><font color="Green">E247</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E11:38_4"><i><font color="Green">E72</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">WTG</font> = <font color="Maroon">WT</font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E8:38_4_1_1_2_5_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">WTG</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>then consider </b><font color="Maroon">WX</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E10:38_4_1_1_2_5_2"/><i><font color="Green">E251</font></i>: 
( <font color="Maroon">WX</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">WX</font> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E11:38_4_1_1_2_5_2"/>
( <font color="Maroon">WX</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">WY</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">WY</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">WY</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, </i>;<br/><a NAME="E12:38_4_1_1_2_5_2"/><b>then </b>
<font color="Maroon">WT</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E13:38_4_1_1_2_5_2"/><b>then </b><i><font color="Green">E253</font></i>: 
<font color="Maroon">WT'</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:38_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="glib_003.html#T34">GLIB_003:34</a></i>;<br/><a NAME="E14:38_4_1_1_2_5_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">WTG</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, </i>;<br/><a NAME="E15:38_4_1_1_2_5_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">WT'</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T34">GLIB_003:34</a></i>;<br/><b>hence </b><a NAME="E16:38_4_1_1_2_5_2"/>
<font color="Maroon">WT'</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:38_4_1_1_2_5"/>
<font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#T32">GLIB_003:32</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">W2</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">Dn1W</font><b> such that </b><br/><a NAME="E48:38_4_1_1_2"/><i><font color="Green">E254</font></i>: 
( <font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">W2G</font> = <font color="Maroon">W2</font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E50:38_4_1_1_2"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">W2G</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E51:38_4_1_1_2"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">W2G</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T34">GLIB_003:34</a></i>;<br/><a NAME="E52:38_4_1_1_2"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">src</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T2" target="_self">Th2</a>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>set </b><font color="Maroon">mc</font> = <font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_6"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">WB</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:38_4_1_1_2_6"/><i><font color="Green">E259</font></i>: 
( <font color="Maroon">WB</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">WB</font> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, </i>;<br/><a NAME="E3:38_4_1_1_2_6"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">WB</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">WA</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">WA</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">WB</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">WA</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E4:38_4_1_1_2_6"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="glib_004.html#E17:38_4"><i><font color="Green">E79</font></i></a>, </i>;<br/><a NAME="E5:38_4_1_1_2_6"/>
( <font color="Maroon">WB</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">src</font> &amp; <font color="Maroon">WB</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E6:38_4_1_1_2_6"/><b>then </b>
not <font color="Maroon">WB</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:38_4_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><b>then consider </b><font color="Maroon">lenWB2h</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:38_4_1_1_2_6"/><i><font color="Green">E262</font></i>: 
( <font color="Maroon">lenWB2h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">WB</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">WB</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T134">GLIB_001:134</a></i>;<br/><b>set </b><font color="Maroon">WA</font> = <font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font>;<br/><b>set </b><font color="Maroon">sa</font> = <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenWB2h</font>;<br/><b>set </b><font color="Maroon">ea</font> = <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E9:38_4_1_1_2_6"/><i><font color="Green">E265</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWB2h</font> &amp; not 1 is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E10:38_4_1_1_2_6"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">lenWB2h</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">WB</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><i><font color="Green">E267</font></i>: <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">WB</font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>
<br/>.= 
<font color="Maroon">src</font>
<b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><a NAME="E12:38_4_1_1_2_6"/><b>then </b>
<font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenWB2h</font>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">sa</font> = <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenWB2h</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T19">GLIB_001:19</a></i>;<br/><b>reconsider </b><font color="Maroon">target'</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_001.html#T19">GLIB_001:19</a></i>;<br/><a NAME="E15:38_4_1_1_2_6"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sa</font>,<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T123">GLIB_001:123</a></i>;<br/><a NAME="E16:38_4_1_1_2_6"/><b>then </b>
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sa</font>,<font color="Maroon">WB</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E17:38_4_1_1_2_6"/><b>then </b><i><font color="Green">E270</font></i>: 
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sa</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E18:38_4_1_1_2_6"/><b>then </b>
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">sa</font>,<font color="Maroon">target'</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><a NAME="E19:38_4_1_1_2_6"/><b>then </b>
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">sa</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T65">GLIB_000:65</a></i>;<br/><a NAME="E20:38_4_1_1_2_6"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K4">.last()</a> </span>)</span> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i>, <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E21:38_4_1_1_2_6"/><b>then </b><i><font color="Green">E272</font></i>: 
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </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">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_003.html#T32">GLIB_003:32</a></i>;<br/><a NAME="E22:38_4_1_1_2_6"/><i><font color="Green">E273</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">src</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">sa</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38_4_1_1_2"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E23:38_4_1_1_2_6"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">src</font>,<font color="Maroon">sa</font>
 <b>by </b><i>, </i>;<br/><a NAME="E24:38_4_1_1_2_6"/><b>then </b><i><font color="Green">E275</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">sa</font>
 <b>by </b><i/>;<br/><a NAME="E25:38_4_1_1_2_6"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sa</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="glib_004.html#E9:38_4_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_4_1_1_2_6_2"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> &amp; not <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  );<br/><div><i><font color="Green">E278</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_4_1_1_2_6_2_1"/><i><font color="Green">E279</font></i>: 
not <font color="Maroon">sa</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><font color="Maroon">sa</font> = 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i>, <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
;<br/><a NAME="E3:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E280</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E20:38_4_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:38_4_1_1_2_6_2_1"/><i><font color="Green">E281</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="glib_004.html#E22:38_4_1_1_2"><i><font color="Green">E98</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>
<br/>.= 
<font color="Maroon">src</font>
<b>by </b><i><a class="txt" href="glib_004.html#E7:38_4_1_1_2"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
;<br/><a NAME="E8:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="glib_004.html#T2" target="_self">Th2</a>, , <a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E9:38_4_1_1_2_6_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E10:38_4_1_1_2_6_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E24:38_4_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E11:38_4_1_1_2_6_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E12:38_4_1_1_2_6_2_1"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">k2a</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon">sk</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2a</font>;<br/><b>set </b><font color="Maroon">ek</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">tk</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/><a NAME="E14:38_4_1_1_2_6_2_1"/><i><font color="Green">E286</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2a</font> &amp; <font color="Maroon">k2a</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E15:38_4_1_1_2_6_2_1"/>
<font color="Maroon">k2a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E16:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E287</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="glib_004.html#E25:38_4_1_1_2"><i><font color="Green">E101</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">sk</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2a</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">tk</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><b>set </b><font color="Maroon">WK1</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font>;<br/><b>set </b><font color="Maroon">WK2</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k2a</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">WKA</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font>;<br/><b>set </b><font color="Maroon">WKB</font> = <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span>;<br/><a NAME="E19:38_4_1_1_2_6_2_1"/>
<font color="Maroon">tk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/><a NAME="E20:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E292</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sk</font>,<font color="Maroon">tk</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E25:38_4_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="glib_001.html#T123">GLIB_001:123</a></i>;<br/><a NAME="E21:38_4_1_1_2_6_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">sk</font>,<font color="Maroon">tk</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E22:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E293</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sk</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tk</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E23:38_4_1_1_2_6_2_1"/><i><font color="Green">E294</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">sk</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E25:38_4_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E24:38_4_1_1_2_6_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">sk</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:38_4_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="glib_004.html#E26:38_4_1_1_2"><i><font color="Green">E105</font></i></a>, </i>;<br/><a NAME="E25:38_4_1_1_2_6_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<font color="Maroon">sk</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>;<br/><a NAME="E26:38_4_1_1_2_6_2_1"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">sk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E12:38_4_1_1_2"><i><font color="Green">E91</font></i></a>, </i>;<br/><a NAME="E27:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E295</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">sk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E28:38_4_1_1_2_6_2_1"/>
<font color="Maroon">tk</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E29:38_4_1_1_2_6_2_1"/><b>then </b>
<font color="Maroon">tk</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E30:38_4_1_1_2_6_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><a NAME="E31:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E296</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </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">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="glib_004.html#E28:38_4_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E5:38"><i><font color="Green">E47</font></i></a>, , </i>;<br/><a NAME="E32:38_4_1_1_2_6_2_1"/>
<font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E33:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E297</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k2a</font></span>)</span> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E25:38_4_1_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="glib_001.html#T42">GLIB_001:42</a></i>;<br/><a NAME="E34:38_4_1_1_2_6_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sk</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tk</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E35:38_4_1_1_2_6_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2a</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">sk</font> <a href="glib_000.html#K29">.edgesInOut()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T64">GLIB_000:64</a></i>;<br/><a NAME="E36:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E298</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E26:38_4_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E29:38_4_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_004.html#E30:38_4_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="glib_003.html#T32">GLIB_003:32</a></i>;<br/><a NAME="E37:38_4_1_1_2_6_2_1"/><i><font color="Green">E299</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a></i>;<br/><b>then </b><i><font color="Green">E300</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40">GLIB_001:40</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="glib_004.html#E23:38_4_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
;<br/><a NAME="E40:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E301</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_003.html#T31">GLIB_003:31</a></i>;<br/><a NAME="E41:38_4_1_1_2_6_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">k</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T36">GLIB_003:36</a></i>;<br/><a NAME="E42:38_4_1_1_2_6_2_1"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E43:38_4_1_1_2_6_2_1"/><b>then </b><i><font color="Green">E302</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E32:38_4_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E44:38_4_1_1_2_6_2_1"/>
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:38_4_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E45:38_4_1_1_2_6_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#T38">GLIB_003:38</a></i>;<br/><a NAME="E46:38_4_1_1_2_6_2_1"/><b>then </b>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </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">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E47:38_4_1_1_2_6_2_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_003.html#T32">GLIB_003:32</a>, <a class="txt" href="glib_004.html#E17:38_4_1_1_2"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:38_4_1_1_2_6_2"/><i><font color="Green">E303</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">lenWB2h</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">sa</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:38_4_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="glib_004.html#E15:38_4_1_1_2"><i><font color="Green">E93</font></i></a>, </i>;<br/><a NAME="E4:38_4_1_1_2_6_2"/><i><font color="Green">E304</font></i>: 
( <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sa</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4_1_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E5:38_4_1_1_2_6_2"/><b>then </b>
<font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lenWB2h</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R4">DSJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:38_4_1_1_2"><i><font color="Green">E96</font></i></a>, , <a class="ref" href="glib_000.html#D18">GLIB_000:def 18</a></i>;<br/><b>hence </b><a NAME="E6:38_4_1_1_2_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E11:38_4_1_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="glib_004.html#E17:38_4_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="glib_004.html#E20:38_4_1_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="glib_004.html#E2:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="glib_004.html#E5:38"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="glib_004.html#E22:38_4_1_1_2"><i><font color="Green">E98</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E27:38_4_1_1_2_6"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E54:38_4_1_1_2"/><b>then </b><i><font color="Green">E305</font></i>: 
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Dn1W</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_1_2_7_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_7_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">DnW</font> <b>by </b><i>, <a class="ref" href="glib_004.html#T3" target="_self">Th3</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:38_4_1_1_2_7_1_1_1"/>
<font color="Maroon">DnW</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">W2</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">DnW</font><b> such that </b><br/><a NAME="E5:38_4_1_1_2_7_1_1_1"/><i><font color="Green">E307</font></i>: 
( <font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x'</font> &amp; ( for <font color="Olive">W1</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x'</font> holds <br/><font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W1</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">W2'</font> = <font color="Maroon">W2</font> as   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">Dn1W</font> <b>by </b><i><a class="txt" href="glib_004.html#E5:38_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><b>take </b><font color="Maroon">W2'</font> = <font color="Maroon">W2'</font>;<br/><b>thus </b><a NAME="E7:38_4_1_1_2_7_1_1_1"/>
<font color="Maroon">W2'</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>let </b><font color="Maroon">W1</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E8:38_4_1_1_2_7_1_1_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E9:38_4_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><b>hence </b><a NAME="E10:38_4_1_1_2_7_1_1_1"/>
<font color="Maroon">W2'</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E10:38_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="glib_003.html#T34">GLIB_003:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_7_1_1_2"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green">E310</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E3:38_4_1_1_2_7_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>let </b><font color="Maroon">W1</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E4:38_4_1_1_2_7_1_1_2"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">WX</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:38_4_1_1_2_7_1_1_2"/><i><font color="Green">E312</font></i>: 
( <font color="Maroon">WX</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">src</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">WX</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E16:38_4"><i><font color="Green">E78</font></i></a>, , , </i>;<br/><b>thus </b><a NAME="E7:38_4_1_1_2_7_1_1_2"/>
<font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:38_4_1_1_2_7"/>
 ex <font color="Olive">W2</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">Dn1W</font> st <br/>( <font color="Olive">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font> &amp; ( for <font color="Olive">W1</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">src</font>,<font color="Maroon">x</font> holds <br/><font color="Olive">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W1</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E56:38_4_1_1_2"/>
<font color="Maroon">Dn1W</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E15:38_4"><i><font color="Green">E76</font></i></a>, </i>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E57:38_4_1_1_2"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_1_2_8_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_8_1_1"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2_8_1_1"/><b>then </b><i><font color="Green">E315</font></i>: 
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:38_4_1_1_2_8_1_1"/><i><font color="Green">E316</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="glib_003.html#T45">GLIB_003:45</a></i>;<br/><a NAME="E4:38_4_1_1_2_8_1_1"/><i><font color="Green">E317</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E5:38_4_1_1_2_8_1_1"/><b>then </b><i><font color="Green">E318</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:38_4_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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#E7:38_4_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="glib_004.html#E2:38_4_1_1_2"><i><font color="Green">E82</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:38_4_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon">v</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span>
 ;<br/><a NAME="E8:38_4_1_1_2_8_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:38_4_1_1_2_8_1_1"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="glib_004.html#E7:38_4_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="glib_004.html#E8:38_4_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_1_2_8_1_2"/><i><font color="Green">E319</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:38_4_1_1_2_8_1_2"/><b>then </b><i><font color="Green">E320</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:38_4_1_1_2_8_1_2"/><i><font color="Green">E321</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:38"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="glib_003.html#T45">GLIB_003:45</a></i>;<br/><a NAME="E4:38_4_1_1_2_8_1_2"/><i><font color="Green">E322</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E5:38_4_1_1_2_8_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:38_4_1_1_2_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></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> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:38_4_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="glib_004.html#E5:38_4_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="glib_004.html#E7:38_4_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E7:38_4_1_1_2_8_1_2"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E59:38_4_1_1_2"/>
<font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:38_4"/>
( <font color="Maroon">Dn1W</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font> &amp; ( for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/><font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">v</font> ) )
 ;<br/></div><b>end;</b></div>
<a NAME="E11:38"/><b>then </b><i><font color="Green">E323</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E12:38"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, );<br/></i>
<b>hence </b><a NAME="E13:38"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">G2</font> being  <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  holds <br/> ( <font color="Olive">G2</font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">src</font> &amp; ( for <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  holds <br/><font color="Maroon">G</font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">src</font>,<font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">v</font> ) )
 ;<br/>


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