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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</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>let </b><font color="Maroon" title="c3">G3</font> be   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font>;<br/>



<b>assume </b><a NAME="E1:84"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c3">G3</font> is   <a href="glib_004.html#NM2" title="GLIB_004:NM.2">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">G2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">G3'</font> = <font color="Maroon" title="c3">G3</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G3'</font> = <font color="Maroon" title="c3">G3</font> as  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_003.html#T17" title="GLIB_003:th.17">GLIB_003:17</a></i>;<br/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G3</font> = 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">G3'</font> = <font color="Maroon" title="c5">G3'</font> as  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">G</font> be  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c2">G2</font>;<br/><b>reconsider </b><font color="Maroon" title="c8">G'</font> = <font color="Maroon" title="c7">G</font> as  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_003.html#T17" title="GLIB_003:th.17">GLIB_003:17</a></i>;<br/> <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c7">G</font> = 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
<b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/><a NAME="E3:84_2"/><b>then </b>
<font color="Maroon" title="c8">G'</font> is <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><b>hence </b><a NAME="E4:84_2"/>
<font color="Maroon" title="c3">G3</font> <a href="glib_004.html#K10" title="GLIB_004:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">G</font> <a href="glib_004.html#K10" title="GLIB_004:func.10">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="glib_004.html#D17" target="_self" title="GLIB_004:def.17">Def17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:84"/><b>then </b>
<font color="Maroon" title="c6">G3'</font> is <a href="glib_004.html#V1" title="GLIB_004:attr.1">min-cost</a>
 
<b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self" title="GLIB_004:def.17">Def17</a></i>;<br/>
<b>hence </b><a NAME="E7:84"/>
<font color="Maroon" title="c3">G3</font> is   <a href="glib_004.html#NM2" title="GLIB_004:NM.2">minimumSpanningTree</a> of <font color="Maroon" title="c2">G2</font>
 ;<br/>


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