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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font> be   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E54</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="glib_002.html#V3">Tree-like</a> <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_003.html#T17">GLIB_003:17</a></i>;<br/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
<br/>.= 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</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>2</sub></font> <b>by </b><i><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="81_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</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>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <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#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_003.html#T17">GLIB_003:17</a></i>;<br/> <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</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="txt" href="glib_004.html#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/><a NAME="E3:81_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="glib_000.html#V9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><b>hence </b><a NAME="E4:81_2"/>
<font color="Maroon">c<sub>3</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>7</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:81"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:81"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="glib_004.html#V1" target="_self">min-cost</a>
 
<b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E7:81"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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