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<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is  <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>1</sub></font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:79_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is  <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>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:79_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  <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>1</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:79_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:79_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is  <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>1</sub></font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> </span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <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>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> ;<br/>
<a NAME="E4:79_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="glib_004.html#E37"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_003.html#T15">GLIB_003:15</a></i>;<br/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E4:79_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</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="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E4:79_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a>, <a class="ref" href="glib_002.html#T48">GLIB_002:48</a></i>;<br/>
<a NAME="E8:79_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> 
 
<b>by </b><i><a class="txt" href="glib_004.html#E38"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E9:79_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 
<b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E10:79_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> </span>)</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <font color="Maroon">a<sub>1</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E13:79_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2">GRAPH_5:sch 2</a>();<br/></i>
<a NAME="E14:79_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E16:79_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is  <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>1</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</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>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E16:79_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</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>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> ;<br/><a NAME="E1:79_1_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E37"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_003.html#T15">GLIB_003:15</a></i>;<br/> <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>14</sub></font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:79_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</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="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</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>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E1:79_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a>, <a class="ref" href="glib_002.html#T48">GLIB_002:48</a></i>;<br/><a NAME="E5:79_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E38"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E6:79_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E7:79_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E8:79_1_1_3"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>11</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>15</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E13:79_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E9:79_1_1_3"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:79_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E10:79_1_1_3"/>
<font color="Maroon">c<sub>11</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>12</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E1:79_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="glib_004.html#E8:79_1_1_3"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:79_1_1"/>
<font color="Maroon">c<sub>11</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/>


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