<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  holds <font color="Olive">b<sub>1</sub></font> is <a href="glib_002.html#V1">connected</a>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E1:66_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0 <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E2:66_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> 0</span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E55"><i><font color="Green">Lemma39</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>, <a href="xboole_0.html#K1">{}</a>  ;<br/><a NAME="E4:66_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="glib_002.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E5:66_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_002.html#V1">connected</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:66"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:66_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self">Def15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E1:66_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E2:66_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_1"/>
(  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or (  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or (  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</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="66_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_1"/>
 <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self">Def14</a></i>;<br/><b>hence </b><a NAME="E3:66_2_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E3:66_2_1_1_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2_1_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:66_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41">GLIB_003:41</a></i>;<br/><a NAME="E6:66_2_1_1_1_2"/><b>then </b>
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:66_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><a NAME="E8:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_003.html#T63">GLIB_003:63</a></i>;<br/><a NAME="E9:66_2_1_1_1_2"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_003.html#T49">GLIB_003:49</a></i>;<br/><a NAME="E10:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><a NAME="E11:66_2_1_1_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:66_2_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_004.html#E3:66_2_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E13:66_2_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E14:66_2_1_1_1_2"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E15:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E16:66_2_1_1_1_2"/><b>then </b>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E8:66_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="glib_004.html#E11:66_2_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><a NAME="E20:66_2_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E21:66_2_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:66_2_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E22:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E20:66_2_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E23:66_2_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E24:66_2_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E11:66_2_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E25:66_2_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E22:66_2_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_2_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="glib_004.html#E14:66_2_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E4:66_2_1_1_1_2_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E13:66_2_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E6:66_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:66_2_1_1_1_2_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E7:66_2_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E25:66_2_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="glib_004.html#E1:66_2_1_1_1_2_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><b>hence </b><a NAME="E8:66_2_1_1_1_2_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E15:66_2_1_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E8:66_2_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E5:66_2_1_1_1_2_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E13:66_2_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E7:66_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:66_2_1_1_1_2_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E8:66_2_1_1_1_2_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:66_2_1_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E27:66_2_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T6">GLIB_002:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_3"/><i><font color="Green">E49</font></i>: 
(  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E3:66_2_1_1_1_3"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R3">SJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2_1_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E4:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2_1_1_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="glib_000.html#D17">GLIB_000:def 17</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:66_2_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41">GLIB_003:41</a></i>;<br/><a NAME="E7:66_2_1_1_1_3"/><b>then </b>
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E8:66_2_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><a NAME="E9:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_003.html#T63">GLIB_003:63</a></i>;<br/><a NAME="E10:66_2_1_1_1_3"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="glib_003.html#T49">GLIB_003:49</a></i>;<br/><a NAME="E11:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><a NAME="E12:66_2_1_1_1_3"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E11:66_2_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="glib_004.html#E4:66_2_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/><a NAME="E14:66_2_1_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E15:66_2_1_1_1_3"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E16:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E57</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E17:66_2_1_1_1_3"/><b>then </b>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E9:66_2_1_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="glib_004.html#E12:66_2_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T45">GLIB_000:45</a></i>;<br/><a NAME="E21:66_2_1_1_1_3"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E22:66_2_1_1_1_3"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:66_2_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E23:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E21:66_2_1_1_1_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E24:66_2_1_1_1_3"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:66_2_1_1_1_3"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:66_2_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E26:66_2_1_1_1_3"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:66_2_1_1_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_3_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E4:66_2_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="glib_004.html#E15:66_2_1_1_1_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E4:66_2_1_1_1_3_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:66_2_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E6:66_2_1_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:66_2_1_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E7:66_2_1_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:66_2_1_1_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="glib_004.html#E1:66_2_1_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><b>hence </b><a NAME="E8:66_2_1_1_1_3_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_3_1_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E16:66_2_1_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_004.html#E2:66_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="glib_004.html#E9:66_2_1_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E5:66_2_1_1_1_3_1_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:66_2_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E7:66_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:66_2_1_1_1_3_1_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E8:66_2_1_1_1_3_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:66_2_1_1_1_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E28:66_2_1_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T6">GLIB_002:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="glib_002.html#V1">connected</a>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:66"/><b>then </b><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:66"/>
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_004.html#E2:66"><i><font color="Green">E45</font></i></a>, <a class="txt" href="glib_004.html#E4:66"><i><font color="Green">E46</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:66"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  holds <font color="Olive">b<sub>2</sub></font> is <a href="glib_002.html#V1">connected</a>
 ;<br/>


</div>
