<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c2">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">G2</font> where <font color="Olive" title="b1">G2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">G2</font> is  <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">G1</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G2</font> where <font color="Olive" title="b1">G2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">G2</font> is  <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">G1</font> </span>}</span>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  )</span><br/><b>assume </b><a NAME="E1:78_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G2</font> where <font color="Olive" title="b1">G2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">G2</font> is  <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">G1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span><br/><b>then consider </b><font color="Maroon" title="c4">G2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:78_1_1_1"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> &amp; <font color="Maroon" title="c4">G2</font> is  <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">G1</font> )
 ;<br/><b>thus </b><a NAME="E4:78_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 <b>by </b><i><a class="txt" href="#E3:78_1_1_1"><i><font color="Green" title="E54">A1</font></i></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>then reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">G2</font> where <font color="Olive" title="b1">G2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">G2</font> is  <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">G1</font> </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">GT</font> being  <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">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">G3</font> = <font color="Maroon" title="c4">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ;<br/>
<a NAME="E4:78_1_1"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c4">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">GT</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> </span>)</span> <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="c4">GT</font> )
 
<b>by </b><i><a class="txt" href="glib_004.html#E17"><i><font color="Green" title="E13">Lm2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">G3</font> = <font color="Maroon" title="c4">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  as   <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="ref" href="glib_003.html#T15" title="GLIB_003:th.15">GLIB_003:15</a></i>;<br/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">G3</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">GT</font>
<b>by </b><i><a class="txt" href="#E4:78_1_1"><i><font color="Green" title="E54">A2</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">G1</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="c7">G3</font> = <font color="Maroon" title="c6">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="c1">G1</font> <b>by </b><i><a class="txt" href="#E4:78_1_1"><i><font color="Green" title="E54">A2</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/>
<a NAME="E8:78_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> 
 
<b>by </b><i><a class="txt" href="glib_004.html#E18"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/>
<a NAME="E9:78_1_1"/><b>then </b>
<font color="Maroon" title="c7">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 
<b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/>
<a NAME="E10:78_1_1"/><b>then </b>
<font color="Maroon" title="c7">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c3">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span>)</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</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>) <b>-&gt; </b>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> ;<br/>
<b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c8">X</font><b> such that </b><br/><a NAME="E13:78_1_1"/><i><font color="Green" title="E55">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c8">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></i>
<a NAME="E14:78_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">G2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E16:78_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">G2</font> &amp; <font color="Maroon" title="c10">G2</font> is  <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">G1</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c9">x</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">G1</font> <b>by </b><i><a class="txt" href="#E16:78_1_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> is <a href="glib_004.html#V1" title="GLIB_004:attr.1">min-cost</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">GT</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">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</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="c12">GT</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span><br/><b>set </b><font color="Maroon" title="c13">G3</font> = <font color="Maroon" title="c12">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ;<br/><a NAME="E1:78_1_1_3"/><i><font color="Green" title="E57">A5</font></i>: 
( <font color="Maroon" title="c12">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c12">GT</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> </span>)</span> <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="c12">GT</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E17"><i><font color="Green" title="E13">Lm2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">G3</font> = <font color="Maroon" title="c12">GT</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  as   <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="ref" href="glib_003.html#T15" title="GLIB_003:th.15">GLIB_003:15</a></i>;<br/> <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">G3</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c12">GT</font>
<b>by </b><i><a class="txt" href="#E1:78_1_1_3"><i><font color="Green" title="E57">A5</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">G1</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="c15">G3</font> = <font color="Maroon" title="c14">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="c1">G1</font> <b>by </b><i><a class="txt" href="#E1:78_1_1_3"><i><font color="Green" title="E57">A5</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/><a NAME="E5:78_1_1_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E18"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/><a NAME="E6:78_1_1_3"/><b>then </b>
<font color="Maroon" title="c15">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E7:78_1_1_3"/><b>then </b>
<font color="Maroon" title="c15">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/><a NAME="E8:78_1_1_3"/><b>then </b>
<font color="Maroon" title="c11">x</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="c15">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="txt" href="#E13:78_1_1"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E9:78_1_1_3"/>
<font color="Maroon" title="c11">x</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="c12">GT</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="txt" href="#E1:78_1_1_3"><i><font color="Green" title="E57">A5</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="E19:78_1_1"/>
<font color="Maroon" title="c11">x</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#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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