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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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">M</font> =  <a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c3">PCS</font> =  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c4">V</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">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">G</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">RFS</font> = <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:79"/>
<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">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E66"><i><font color="Green" title="E50">Lm16</font></i></a></i>;<br/>
<a NAME="E2:79"/><b>then </b><i><font color="Green" title="E53">A1</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</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>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E65"><i><font color="Green" title="E49">Lm15</font></i></a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E54">A2</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E62"><i><font color="Green" title="E46">Lm12</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:79_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</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_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:79"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E3:79_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E2:79"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:79"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</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>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_002.html#T16" title="GLIB_002:th.16">GLIB_002:16</a></i>;<br/>


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