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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>; <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">G1</font> being  <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">G2</font> being  <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>  st <font color="Olive" title="b1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b2">G2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Olive" title="b1">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="Olive" title="b2">G2</font> holds <br/><font color="Olive" title="b2">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font></span><br/><b>let </b><font color="Maroon" title="c2">G1</font> be   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</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"> for <font color="Olive" title="b1">G2</font> being  <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>  st <font color="Maroon" title="c2">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b1">G2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">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="Olive" title="b1">G2</font> holds <br/><font color="Olive" title="b1">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font></span><br/><b>let </b><font color="Maroon" title="c3">G2</font> be   <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"> ( <font color="Maroon" title="c2">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">G2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">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="c3">G2</font> implies <font color="Maroon" title="c3">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font> )</span><br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E55">A1</font></i>: 
( <font color="Maroon" title="c2">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">G2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c2">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="c3">G2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">G2'</font> = <font color="Maroon" title="c3">G2</font> as   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_003.html#T15" title="GLIB_003:th.15">GLIB_003:15</a></i>;<br/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font>
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E55">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 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_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">G2'</font> = <font color="Maroon" title="c4">G2'</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="c1">G</font> <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a>, <a class="ref" href="glib_002.html#T48" title="GLIB_002:th.48">GLIB_002:48</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" title="c6">G3</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="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="c5">G2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span><br/><a NAME="E1:81_2"/>
<font color="Maroon" title="c2">G1</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D19" target="_self" title="GLIB_004:def.19">Def17</a></i>;<br/><b>hence </b><a NAME="E2:81_2"/>
<font color="Maroon" title="c5">G2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</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>
<b>hence </b><a NAME="E6:81"/>
<font color="Maroon" title="c3">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_004.html#D19" target="_self" title="GLIB_004:def.19">Def17</a></i>; <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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