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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">src</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">G2</font> being  <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">src</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  holds <br/> ( <font color="Olive" title="b2">G2</font> <a href="glib_004.html#R1" title="GLIB_004:pred.1">is_mincost_DTree_rooted_at</a> <font color="Olive" title="b1">src</font> &amp; ( for <font color="Olive" title="b3">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Olive" title="b1">src</font> holds <br/>( <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Olive" title="b2">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Olive" title="b1">src</font>,<font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">v</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">src</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being  <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  holds <br/> ( <font color="Olive" title="b1">G2</font> <a href="glib_004.html#R1" title="GLIB_004:pred.1">is_mincost_DTree_rooted_at</a> <font color="Maroon" title="c2">src</font> &amp; ( for <font color="Olive" title="b2">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> holds <br/>( <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Olive" title="b1">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">v</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">G2</font> be   <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">G2</font> <a href="glib_004.html#R1" title="GLIB_004:pred.1">is_mincost_DTree_rooted_at</a> <font color="Maroon" title="c2">src</font> &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> holds <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> ) ) )</span><br/>





<b>set </b><font color="Maroon" title="c4">Res</font> =  <a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>;<br/>
<b>thus </b><a NAME="E1:43"/>
<font color="Maroon" title="c3">G2</font> <a href="glib_004.html#R1" title="GLIB_004:pred.1">is_mincost_DTree_rooted_at</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T22" target="_self" title="GLIB_004:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> holds <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> implies ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> ) )</span><br/>

<b>assume </b><a NAME="E2:43"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> )</span><br/>

<a NAME="E3:43"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 
<b>by </b><i><a class="ref" href="glib_004.html#T25" target="_self" title="GLIB_004:th.25">Th25</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#T17" target="_self" title="GLIB_004:th.17">Th17</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E2:43"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="glib_004.html#T25" target="_self" title="GLIB_004:th.25">Th25</a></i>;<br/>
<a NAME="E6:43"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E4:43"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="glib_000.html#T93" title="GLIB_000:th.93">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" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T21" target="_self" title="GLIB_004:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:43"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="glib_004.html#E5:43"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font></span><br/>

<b>thus </b><a NAME="E9:43"/>
<font color="Maroon" title="c1">G</font> <a href="glib_004.html#K1" title="GLIB_004:func.1">.min_DPath_cost</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:43"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="glib_004.html#T22" target="_self" title="GLIB_004:th.22">Th22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
