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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>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<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>set </b><font color="Maroon">dDCS</font> = <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">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<br/>
<b>set </b><font color="Maroon">eDCS</font> = <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">i</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>
<b>assume </b><a NAME="E1:31"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<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>( <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">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <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">i</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  );<br/>
<a NAME="E6:31"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><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="E1:31_1"/><i><font color="Green">E48</font></i>: 
( <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">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <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">i</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 ;<br/><a NAME="E2:31_1"/>
<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"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:31_1"/><b>then </b>
( <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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</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"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <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">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</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"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:31_1"/>
( <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">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <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">i</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</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">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:31"/><b>then </b><i><font color="Green">E49</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="E9:31"/>
for <font color="Olive">k</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">k</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="E10:31"/>
( <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">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</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">j</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <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">i</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</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">j</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i/>;<br/>


</div>
