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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_003.html#V11" title="GLIB_003:attr.11">real-WEV</a> <a href="glib_003.html#NM7" title="GLIB_003:NM.7">WEVGraph</a>;<br/>

<b>set </b><font color="Maroon" title="c2">G2</font> =  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">Next</font> =  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">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> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">GE</font> = <font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_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_1_1"/>
(  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font> <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="66_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1"/>
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <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"><b>hence </b><a NAME="E2:66_1_1_1"/>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_2"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c1">G</font> <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="c6">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:66_1_1_2"/><i><font color="Green" title="E43">A2</font></i>: 
( not <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp;  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <font color="Maroon" title="c6">v</font>,1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E56"><i><font color="Green" title="E40">Lm6</font></i></a></i>;<br/><a NAME="E4:66_1_1_2"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></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="E5:66_1_1_2"/>
<font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:66_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="glib_003.html#T47" title="GLIB_003:th.47">GLIB_003:47</a></i>;<br/><b>hence </b><a NAME="E6:66_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:66_1_1_2"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="glib_000.html#T88" title="GLIB_000:th.88">GLIB_000:88</a></i>;<br/><a NAME="E7:66_1_1_2"/>
 <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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:66_1_1_2"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self" title="GLIB_004:def.12">Def12</a></i>;<br/><a NAME="E8:66_1_1_2"/><b>then </b><i><font color="Green" title="E45">A4</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> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>;<br/><b>thus </b> <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> = 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#T42" title="GLIB_003:th.42">GLIB_003:42</a></i>
<br/>.= 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E3:66_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="glib_003.html#T48" title="GLIB_003:th.48">GLIB_003:48</a></i>
;<br/><a NAME="E10:66_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E8:66_1_1_2"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="glib_003.html#T53" title="GLIB_003:th.53">GLIB_003:53</a></i>;<br/><b>hence </b><a NAME="E11:66_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:66_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="glib_003.html#T56" title="GLIB_003:th.56">GLIB_003:56</a></i>;<br/><a NAME="E12:66_1_1_2"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <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"><font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:66_1_1_2"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><a NAME="E13:66_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</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#E3:66_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="glib_003.html#T60" title="GLIB_003:th.60">GLIB_003:60</a></i>;<br/><b>hence </b><a NAME="E14:66_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:66"/>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  )
 ;<br/>


</div>
