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<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>let </b><font color="Maroon">G2</font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>





<b>set </b><font color="Maroon">Res</font> =  <a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</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>thus </b><a NAME="E1:43"/>
<font color="Maroon">G2</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>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="E2:43"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">src</font>
 ;<br/>

<a NAME="E3:43"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:43"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</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>, </i>;<br/>
<a NAME="E6:43"/>
<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"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</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#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</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#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T93">GLIB_000:93</a></i>;<br/>
<a NAME="E7:43"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</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"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>

<b>thus </b><a NAME="E9:43"/>
<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"><a href="glib_004.html#K7" target="_self">DIJK:SSSP</a> <font color="Maroon">G</font>,<font color="Maroon">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, </i>;<br/>


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