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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#V7">real-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>set </b><font color="Maroon">c<sub>3</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>4</sub></font> = <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/>
<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/>
<b>set </b><font color="Maroon">c<sub>6</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#K40">.Lifespan()</a> ;<br/>
<a NAME="E1:39"/><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> <a href="hidden.html#R1">=</a> <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>then </b><i><font color="Green">E30</font></i>:  <a href="card_1.html#K4">card</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="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T24" target="_self">Th24</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_002.html#K2">.reachableDFrom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/><i><font color="Green">E31</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>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E2:39_2"/>
<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>  <a href="tarski.html#R1">c=</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>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="glib_004.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:39_2"/><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#K14">.labeledV()</a>  <a href="xboole_0.html#R2">c&lt;</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>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:39_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:39_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E2:39"><i><font color="Green">E30</font></i></a>, <a class="ref" href="trees_1.html#T24">TREES_1:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<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>  <a href="hidden.html#R1">=</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/>


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