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

<b>let </b><font color="Maroon" title="c1">G1</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>;<br/>

<b>set </b><font color="Maroon" title="c2">PCS</font> =  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> $1</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> $1</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  holds  <font color="Olive" title="b1">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>;<br/>
<b>set </b><font color="Maroon" title="c3">G0</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0;<br/>
<b>set </b><font color="Maroon" title="c4">src</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span>;<br/>
<a NAME="E1:78"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:Init</a> <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def15</a></i>;<br/>
<a NAME="E2:78"/><b>then </b><i><font color="Green" title="E52">A1</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="glib_004.html#E55"><i><font color="Green" title="E39">Lm5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">G2</font> be    <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> 0</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> ;<br/><b>reconsider </b><font color="Maroon" title="c6">G2'</font> = <font color="Maroon" title="c5">G2</font> as    <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <b>by </b><i><a class="txt" href="glib_004.html#E2:78"><i><font color="Green" title="E52">A1</font></i></a></i>;<br/><a NAME="E2:78_1"/>
<font color="Maroon" title="c6">G2'</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 ;<br/><b>hence </b><a NAME="E3:78_1"/>
<font color="Maroon" title="c5">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:78"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:78_2"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ;<br/><b>set </b><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font>;<br/><b>set </b><font color="Maroon" title="c7">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">Next</font> =  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>;<br/><a NAME="E2:78_2"/><i><font color="Green" title="E55">A4</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def15</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">G3</font> being    <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> ;<br/><a NAME="E4:78_2"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c10">G3</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:78_2"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/><a NAME="E5:78_2"/><i><font color="Green" title="E57">A6</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green" title="E43">Lm9</font></i></a></i>;<br/><a NAME="E6:78_2"/><b>then </b><i><font color="Green" title="E58">A7</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c10">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c10">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E7:78_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green" title="E42">Lm8</font></i></a></i>;<br/><a NAME="E8:78_2"/><b>then </b><i><font color="Green" title="E59">A8</font></i>: 
(  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">G2</font> be    <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> ;<br/><a NAME="E1:78_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green" title="E43">Lm9</font></i></a></i>;<br/><a NAME="E2:78_2_1"/><b>then </b><i><font color="Green" title="E60">A9</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c11">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c11">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  )
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:78_2_1"/><b>then </b>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c10">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c11">G2</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c10">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c11">G2</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:78_2"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="glib_004.html#E6:78_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="glib_004.html#E57"><i><font color="Green" title="E41">Lm7</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">G3'</font> = <font color="Maroon" title="c10">G3</font> as    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c11">G2</font> <b>by </b><i><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></i>;<br/><a NAME="E5:78_2_1"/>
<font color="Maroon" title="c10">G3</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G3</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E4:78_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>;<br/><a NAME="E6:78_2_1"/><b>then </b><i><font color="Green" title="E61">A10</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E6:78_2"><i><font color="Green" title="E58">A7</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_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="E1:78_2_1_1_1"/>
(  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:78_2_1_1_1_1"/>
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:78_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:78_2"><i><font color="Green" title="E55">A4</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:78_2_1_1_1_1"/>
<font color="Maroon" title="c11">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:78_2"><i><font color="Green" title="E54">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:78_2_1_1_1_2"/><i><font color="Green" title="E62">A11</font></i>: 
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span><b> such that </b><br/><a NAME="E3:78_2_1_1_1_2"/><i><font color="Green" title="E63">A12</font></i>: 
( not <font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <font color="Maroon" title="c13">v</font>,1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:78_2"><i><font color="Green" title="E55">A4</font></i></a>, <a class="txt" href="glib_004.html#E56"><i><font color="Green" title="E40">Lm6</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c14">GnV</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ;<br/><b>set </b><font color="Maroon" title="c15">GnVg</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>;<br/><a NAME="E4:78_2_1_1_1_2"/><i><font color="Green" title="E64">A13</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:78_2_1_1_1_2"><i><font color="Green" title="E62">A11</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self" title="GLIB_004:def.12">Def12</a></i>;<br/><a NAME="E5:78_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E65">A14</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2_1_1_1_2_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 ;<br/><a NAME="E2:78_2_1_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E5:78_2"><i><font color="Green" title="E57">A6</font></i></a>, <a class="txt" href="glib_004.html#E8:78_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="glib_000.html#T34" title="GLIB_000:th.34">GLIB_000:34</a></i>;<br/><a NAME="E3:78_2_1_1_1_2_1"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:78_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E4:78_2_1_1_1_2"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>;<br/></div><b>end;</b></div><a NAME="E7:78_2_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</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#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E5:78_2_1_1_1_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="glib_003.html#T54" title="GLIB_003:th.54">GLIB_003:54</a></i>;<br/><a NAME="E8:78_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A15</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E3:78_2_1_1_1_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="glib_003.html#T56" title="GLIB_003:th.56">GLIB_003:56</a></i>;<br/><a NAME="E9:78_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41" title="GLIB_003:th.41">GLIB_003:41</a></i>;<br/><a NAME="E10:78_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E67">A16</font></i>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">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#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E11:78_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T63" title="GLIB_003:th.63">GLIB_003:63</a></i>;<br/><a NAME="E12:78_2_1_1_1_2"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E3:78_2_1_1_1_2"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="glib_004.html#E10:78_2_1_1_1_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="ref" href="glib_003.html#T61" title="GLIB_003:th.61">GLIB_003:61</a></i>;<br/><a NAME="E13:78_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E2:78_2_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="glib_004.html#E6:78_2_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="txt" href="glib_004.html#E8:78_2_1_1_1_2"><i><font color="Green" title="E66">A15</font></i></a></i>;<br/><a NAME="E14:78_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E68">A17</font></i>: 
<font color="Maroon" title="c11">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E2:78_2_1"><i><font color="Green" title="E60">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E15:78_2_1_1_1_2"/>
<font color="Maroon" title="c11">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E14:78_2_1_1_1_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="glib_002.html#T47" title="GLIB_002:th.47">GLIB_002:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:78_2_1"/>
<font color="Maroon" title="c11">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:78_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:78"/><b>then </b><i><font color="Green" title="E54">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E7:78"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_004.html#E4:78"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="glib_004.html#E6:78"><i><font color="Green" title="E54">A18</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:78"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">G2</font> being   <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  holds  <font color="Olive" title="b2">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 ;<br/>


</div>
