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

<span class="kw">set </span><font color="Maroon" title="c2">GT</font> =  the   <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/>
<span class="kw">set </span><font color="Maroon" title="c3">G3</font> =  the   <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 href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>;<br/>
<a NAME="E1:78_1_1"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 the   <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 href="relat_1.html#K7" title="RELAT_1:func.7">|</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>  the   <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 class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E17"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/>
<a NAME="E2:78_1_1"/>
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"> the   <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 href="relat_1.html#K7" title="RELAT_1:func.7">|</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>  the   <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 class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E17"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">G3</font> =  the   <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 href="relat_1.html#K7" title="RELAT_1:func.7">|</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="glib_003.html#T8" title="GLIB_003:th.8">GLIB_003:8</a></span>;<br/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">G3</font> = 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">G3</font> = <font color="Maroon" title="c4">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a>, <a class="ref" href="glib_002.html#T48" title="GLIB_002:th.48">GLIB_002:48</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">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_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c7">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/><span class="kw">assume </span><a NAME="E1:78_1_1_2"/>
<font color="Maroon" title="c7">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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/><a NAME="E2:78_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G2</font> &amp; <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> )
 ;<br/><span class="kw">hence </span><a NAME="E3:78_1_1_2"/>
<font color="Maroon" title="c7">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>  <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/>
<a NAME="E8:78_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E18"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>;<br/>
<a NAME="E9:78_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def5</a></span>;<br/>
<a NAME="E10:78_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E13:78_1_1"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
 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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></span>
<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/>
<a NAME="E15:78_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G2</font> &amp; <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> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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="c11">GT</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span><br/><span class="kw">set </span><font color="Maroon" title="c12">G3</font> = <font color="Maroon" title="c11">GT</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>;<br/><a NAME="E1:78_1_1_3"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c11">GT</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c11">GT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E17"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/><a NAME="E2:78_1_1_3"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <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="c11">GT</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="c11">GT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E17"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">G3</font> = <font color="Maroon" title="c11">GT</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="glib_003.html#T8" title="GLIB_003:th.8">GLIB_003:8</a></span>;<br/> <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c13">G3</font> = 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c11">GT</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">G3</font> = <font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a>, <a class="ref" href="glib_002.html#T48" title="GLIB_002:th.48">GLIB_002:48</a></span>;<br/><a NAME="E6:78_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_004.html#E18"><span class="lab"><font color="Green" title="E14">Lm5</font></span></a></span>;<br/><a NAME="E7:78_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def5</a></span>;<br/><a NAME="E8:78_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">G3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/><a NAME="E9:78_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c14">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:78_1_1"><span class="lab"><font color="Green" title="E54">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:78_1_1_3"/>
<font color="Maroon" title="c10">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="c11">GT</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_3"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E2:78_1_1_3"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:78_1_1"/>
<font color="Maroon" title="c10">x</font> is  <a href="glib_004.html#V1" title="GLIB_004:attr.1">min-cost</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D19" target="_self" title="GLIB_004:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
