<?xml version="1.0"?>
<div class="add">

<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>, <font color="Maroon">c<sub>3</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>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>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E1:72"/>
<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>2</sub></font>,<font color="Maroon">c<sub>3</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="E2:72"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a>, <a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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>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>2</sub></font>,<font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</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>assume </b><a NAME="E1:72_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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>2</sub></font>,<font color="Maroon">c<sub>3</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> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D23" target="_self">Def23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_1"><b>suppose </b></a><a NAME="E1:72_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E2:72_1_1_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>2</sub></font>,<font color="Maroon">c<sub>3</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#K12" target="_self">FF:AugmentPath</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_005.html#K7" target="_self">AP:GetAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:72_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="glib_005.html#E1:72_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E3:72_1_1_1_1"/><b>then </b>
( <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>2</sub></font>,<font color="Maroon">c<sub>3</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="glib_000.html#R5">==</a> <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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_005.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:72_1_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#E1:72_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_2"><b>suppose </b></a><a NAME="E1:72_1_1_1_2"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <span class="p2">(<span class="default"><span class="p3">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:72_1_1_1_2"/>
<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#E1:72_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="glib_005.html#E3:72_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="glib_005.html#D22" target="_self">Def22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:72_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]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:72"/>
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#E2:72"><i><font color="Green">E38</font></i></a>, <a class="txt" href="glib_005.html#E3:72"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:72"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 ;<br/>


</div>
