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<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font> is <a href="glib_005.html#V4" target="_self">natural-elabeled</a> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font> is <a href="glib_005.html#V3" target="_self">natural-weighted</a> );<br/>
<a NAME="E1:65_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3">ELabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E2:65_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:65_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> as  <a href="glib_000.html#V2">finite</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_005.html#V3" target="_self">natural-weighted</a> <a href="glib_005.html#V4" target="_self">natural-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a> <b>by </b><i><a class="txt" href="glib_005.html#E1:65_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:65_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K13" target="_self">FF:Step</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self">Def23</a></i>;<br/><b>thus </b><a NAME="E5:65_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E4:65_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:65_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="glib_005.html#E2:65_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="glib_005.html#E3:65_1_1"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:65_1_1"/>
(  <a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="glib_005.html#V5" target="_self">natural-weighted</a> &amp;  <a href="glib_005.html#K14" target="_self">FF:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="glib_005.html#V6" target="_self">natural-elabeled</a> )
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a></i>;<br/>


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