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<div class="add">

<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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><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> <font color="Maroon">a<sub>1</sub></font> is <a href="glib_003.html#V8">nonnegative-weighted</a>;<br/>
<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> 0 = 
 <a href="glib_004.html#K4" target="_self">DIJK:Init</a> <font color="Maroon">G</font>,<font color="Maroon">src</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<span class="p1">(<span class="default"><font color="Maroon">src</font> <a href="cqc_lang.html#K3">.--&gt;</a> 0</span>)</span>

;<br/>
<a NAME="E2:22_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then reconsider </b><font color="Maroon">Gn</font> = <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> <font color="Maroon">n</font> as  <a href="glib_003.html#V8">nonnegative-weighted</a> <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a> ;<br/><a NAME="E3:22_1_1_2"/>
<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"><font color="Maroon">n</font> <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> <font color="Maroon">Gn</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:22_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:22_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E5:22_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:22_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:22_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_1_1"/>
 <a href="glib_004.html#K6" target="_self">DIJK:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">src</font> is <a href="glib_003.html#V16">nonnegative-weighted</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D28">GLIB_003:def 28</a></i>;<br/>


</div>
