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<b>let </b><font color="Maroon" title="c1">G</font> be  <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being  <a href="glib_004.html#NM4" title="GLIB_004:NM.4">PRIM:Labeling</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Olive" title="b1">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <a href="glib_004.html#NM4" title="GLIB_004:NM.4">PRIM:Labeling</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )</span><br/>



<b>set </b><font color="Maroon" title="c3">G2</font> =  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font>;<br/>
<b>set </b><font color="Maroon" title="c4">Next</font> =  <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">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#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">GE</font> = <font color="Maroon" title="c2">L</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</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#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</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="63_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_1_1"/>
 <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )</span><br/><div class="add"><b>hence </b><a NAME="E2:63_1_1_1"/>
( <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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"/>
 <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">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="E3:63_1_1_2"/><i><font color="Green" title="E43">A2</font></i>: 
( not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="glib_004.html#T28" target="_self" title="GLIB_004:th.28">Lm6</a></i>;<br/><a NAME="E4:63_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:63_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E5:63_1_1_2"/>
<font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span><br/><a NAME="E6:63_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E3:63_1_1_2"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E7:63_1_1_2"/>
<font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c2">L</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c2">L</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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