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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_003.html#V11">real-WEV</a> <a href="glib_003.html#NM7">WEVGraph</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1"/>
(  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1"/>
 <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <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#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:63_1_1_2"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  &amp;  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">c<sub>8</sub></font>,1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E56"><i><font color="Green">Lemma40</font></i></a></i>;<br/><a NAME="E4:63_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1
 <b>by </b><i><a class="ref" href="glib_003.html#T41">GLIB_003:41</a></i>;<br/><a NAME="E5:63_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1 <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:63_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_003.html#T47">GLIB_003:47</a></i>;<br/><b>hence </b><a NAME="E6:63_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:63_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E7:63_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:63_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E8:63_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#D17">GLIB_000:def 17</a></i>;<br/><b>thus </b> <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#T42">GLIB_003:42</a></i>
<br/>.= 
 <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E3:63_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_003.html#T48">GLIB_003:48</a></i>
;<br/><a NAME="E10:63_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E8:63_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="glib_003.html#T53">GLIB_003:53</a></i>;<br/><b>hence </b><a NAME="E11:63_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <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#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:63_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_003.html#T56">GLIB_003:56</a></i>;<br/><a NAME="E12:63_1_1_2"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:63_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E13:63_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:63_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="glib_003.html#T60">GLIB_003:60</a></i>;<br/><b>hence </b><a NAME="E14:63_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T63">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="E2:63"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <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#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Step</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/>


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