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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V8">nonnegative-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<i><font color="Green">E29</font></i>:  <a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K41">.Result()</a> 

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K5" target="_self">DIJK:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

;<br/>
<b>hence </b><a NAME="E2:40"/>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_004.html#R1" target="_self">is_mincost_DTree_rooted_at</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_004.html#T22" target="_self">Th22</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:40"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:40"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 
<b>by </b><i><a class="ref" href="glib_004.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:40"><i><font color="Green">E29</font></i></a>, <a class="ref" href="glib_004.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:40"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E3:40"><i><font color="Green">E30</font></i></a>, <a class="ref" href="glib_004.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E7:40"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E5:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/>
<a NAME="E8:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:40"><i><font color="Green">E29</font></i></a>, <a class="ref" href="glib_004.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E9:40"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="glib_004.html#E6:40"><i><font color="Green">E33</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>

<b>thus </b><a NAME="E10:40"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K1" target="_self">.min_DPath_cost</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></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"><a href="glib_004.html#K6" target="_self">DIJK:SSSP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:40"><i><font color="Green">E29</font></i></a>, <a class="txt" href="glib_004.html#E4:40"><i><font color="Green">E31</font></i></a>, <a class="ref" href="glib_004.html#T22" target="_self">Th22</a></i>;<br/>


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