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<b>let </b><font color="Maroon">G</font> be  <a href="glib_003.html#V7">real-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>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>( <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">a<sub>1</sub></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">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E1:32"/>
<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="hidden.html#R1">=</a>  <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/>
<a NAME="E2:32"/><b>then </b><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="hidden.html#R1">=</a> <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> ,<a href="xboole_0.html#K1">{}</a> </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="E3:32"/><b>then </b><i><font color="Green">E35</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_000.html#R5">==</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a></i>;<br/>
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> = 
 <a href="glib_003.html#K5">the_Weight_of</a> <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> ,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
 <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> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, , <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E49</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">k</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">k</font></span>)</span> )
 ;<br/><a NAME="E2:32_2"/><i><font color="Green">E50</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">k</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">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:32_2"/><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> <font color="Maroon">k</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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:32_2"/>
<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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>thus </b><a NAME="E5:32_2"/>
 <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">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E8:32"/><b>then </b><i><font color="Green">E51</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="E9:32"/>
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>(, );<br/></i>
<b>hence </b><a NAME="E10:32"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <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="Olive">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="Olive">n</font></span>)</span> )
 ;<br/>


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