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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V1">connected</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_003.html#NM1">WGraph</a>;<br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_003.html#T15">GLIB_003:15</a></i>;<br/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="glib_000.html#V9">spanning</a> <a href="glib_002.html#V3">Tree-like</a> <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_002.html#T48">GLIB_002:48</a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="glib_000.html#V9">spanning</a> <a href="glib_002.html#V3">Tree-like</a> <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:82_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/><a NAME="E2:82_2"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E3:82_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:82_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_004.html#E1:82"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:82"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/>


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