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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V1">connected</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">PCS</font> =  <a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">PMST</font> =  <a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:86"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:86_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">off</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">Next</font> =  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:86_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K13" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:86_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_1_1"/>
 <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:86_1_1"/><i><font color="Green">E50</font></i>: 
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></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#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">v</font>,1 )
 <b>by </b><i>, </i>;<br/><a NAME="E4:86_1_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E5:86_1_1"/><b>then </b>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <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"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</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" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E6:86_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:86_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:86"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  or <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_1"><b>suppose </b></a><a NAME="E1:86_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:86_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_2"><b>suppose </b></a><a NAME="E1:86_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:86_2_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:86_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:86_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:86"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/>


</div>
