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<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>assume </b><a NAME="E1:65"/><i><font color="Green" title="E41">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/>

<b>set </b><font color="Maroon" title="c3">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="c4">src</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">tar</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:65"/><i><font color="Green" title="E42">A2</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_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:65"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>
<a NAME="E3:65"/><i><font color="Green" title="E43">A3</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="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#E1:65"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self" title="GLIB_004:def.12">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">src</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>, <font color="Maroon" title="c7">tar</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a>, <a class="txt" href="glib_004.html#E2:65"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1"/>
( <font color="Maroon" title="c6">src</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>  or  not <font color="Maroon" title="c6">src</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c6">src</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> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">tar</font> = <font color="Maroon" title="c7">tar</font>;<br/><a NAME="E2:65_1_1_1"/>
not <font color="Maroon" title="c6">src</font> <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> <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>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:65_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:65_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">tar</font> <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> <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>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:65"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_1"/>
not <font color="Maroon" title="c8">tar</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> 
 <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>thus </b><a NAME="E5:65_1_1_1"/>
 <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="c8">tar</font>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E1:65_1_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="glib_004.html#E1:65"><i><font color="Green" title="E41">A1</font></i></a>, <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="65_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_2"/><i><font color="Green" title="E44">A5</font></i>: 
not <font color="Maroon" title="c6">src</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> 
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">src</font> = <font color="Maroon" title="c6">src</font>;<br/><b>thus </b><a NAME="E2:65_1_1_2"/>
not <font color="Maroon" title="c8">src</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> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:65_1_1_2"><i><font color="Green" title="E44">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E3:65_1_1_2"/>
 <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="c8">src</font>,1
 <b>by </b><i><a class="txt" href="glib_004.html#E1:65_1_1_2"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="glib_004.html#E1:65"><i><font color="Green" title="E41">A1</font></i></a>, <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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:65"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> st <br/>( not <font color="Olive" title="b1">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="Olive" title="b1">v</font>,1 )
 ;<br/>


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