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

<b>let </b><font color="Maroon">G1</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V1">connected</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/>

<b>set </b><font color="Maroon">PMST</font> =  <a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font>;<br/>
<b>let </b><font color="Maroon">G2</font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>

<b>set </b><font color="Maroon">PCS</font> =  <a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font>;<br/>
<a NAME="E1:87"/>
(  <a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G1</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:87"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E3:87"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:87"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:87"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/>
<a NAME="E6:87"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:87"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 
<b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E8:87"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">G2</font> is <a href="glib_002.html#V3">Tree-like</a> &amp; <font color="Maroon">G2</font> is <a href="glib_000.html#V9">spanning</a> )
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G2'</font> = <font color="Maroon">G2</font> as  <a href="glib_002.html#V3">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G2</font> is  not   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font>
 ;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a>  : <font color="Olive">x</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a>  : <font color="Olive">x</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">G2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:87_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> )
 ;<br/><b>thus </b><a NAME="E4:87_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a>  : <font color="Olive">x</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</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">G1</font> <a href="glib_004.html#K9" 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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">M</font> being   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font>;<br/><b>set </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a> ;<br/><a NAME="E2:87_1_2"/>
( <font color="Maroon">M</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">M</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" 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">M</font> )
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a></i>;<br/><b>then reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><a NAME="E4:87_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a> 
 <b>by </b><i/>;<br/><a NAME="E5:87_1_2"/><b>then </b>
<font color="Maroon">M'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i/>;<br/><a NAME="E6:87_1_2"/><b>then </b>
<font color="Maroon">M'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E7:87_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</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">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> </span>)</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>, <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>] <b>means </b>(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Olive">k2</font>] holds <br/><font color="Olive">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k2</font> ) ) );<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:87_1_3"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">G'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:87_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G'</font> &amp; <font color="Maroon">G'</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><a NAME="E5:87_1_3"/><i><font color="Green">E54</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G'</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_3_1"/><i><font color="Green">E56</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 ;<br/><a NAME="E2:87_1_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G'</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E13:87_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="glib_000.html#T89">GLIB_000:89</a></i>;<br/> <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G2</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
<br/>.= 
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G'</font>
<b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:87_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E10:87_1_3"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="glib_004.html#E5:87_1_3"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_3_2"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 ;<br/><a NAME="E2:87_1_3_2"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:87_1_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:87_1_3_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><a NAME="E4:87_1_3_2"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font></span>)</span>
 ;<br/><a NAME="E5:87_1_3_2"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">G'</font> <a href="glib_000.html#K18">.size()</a> 
 ;<br/><a NAME="E6:87_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:87_1_3_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G'</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/><a NAME="E8:87_1_3_2"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G'</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E13:87_1_3"><i><font color="Green">E64</font></i></a></i>;<br/><b>hence </b><a NAME="E9:87_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E8:87_1_3"/><b>then </b><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:87_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k3</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_3_3"/>
<font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:87_1_3_3"/><b>then </b><i><font color="Green">E63</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:87_1_3_3"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E4:87_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>hence </b><a NAME="E5:87_1_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">k2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:87_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">k2</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k3</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E15:87_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k3</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E16:87_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:87_1_3_4"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font>
 ;<br/><a NAME="E2:87_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E5:87"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E3:87_1_3_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G'</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:87_1_3"/>
 ex <font color="Olive">k1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">G</font>,<font color="Olive">k1</font>]
 <b>by </b><i>, <a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:87_1_4"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">G'</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:87_1_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G'</font> &amp; <font color="Maroon">G'</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E5:87_1_4"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">G</font>,<font color="Maroon">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">G</font>,<font color="Maroon">k2</font>] )
 ;<br/><a NAME="E6:87_1_4"/><b>then </b>
( <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font> )
 ;<br/><a NAME="E7:87_1_4"/><b>then </b>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:87_1_4"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><a NAME="E8:87_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:87_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:87_1_5"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:87_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">y'</font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E6:87_1_5"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">CX</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">CY</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_5_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_5_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_5_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 <b>by </b><i>, <a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_5_1_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:87_1_5_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">k1</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:87_1_5_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">k2</font>]
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_5_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_5_1_1_2_1_1"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font> or <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_5_1_1_2_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_5_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">z2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:87_1_5_1_1_2_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z2</font>] )
 ;<br/><a NAME="E2:87_1_5_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E3:87_1_5"><i><font color="Green">E69</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:87_1_5_1_1_2_1_1_1_1"/>
<font color="Maroon">z1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:87_1_5"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:87_1_5_1_1_2_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 <b>by </b><i>, , <a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_5_1_1_2_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_5_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">z2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:87_1_5_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z2</font>] )
 ;<br/><a NAME="E2:87_1_5_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E3:87_1_5"><i><font color="Green">E69</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:87_1_5_1_1_2_1_1_2_1"/>
<font color="Maroon">z1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:87_1_5"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:87_1_5_1_1_2_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 <b>by </b><i>, , <a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:87_1_5_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:87_1_5_1_1_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_5_1_1_3"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 <b>by </b><i>, <a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:87_1_5"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>] )
 ;<br/></div><b>end;</b></div><a NAME="E10:87_1"/><b>then </b><i><font color="Green">E76</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>  holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:87_1_6"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>] )
 ;<br/><a NAME="E2:87_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:87_1_6"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  )
 <b>by </b><i/>;<br/><a NAME="E5:87_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">y'</font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E7:87_1_6"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  )
 <b>by </b><i/>;<br/><a NAME="E8:87_1_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">z'</font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E10:87_1_6"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">CX</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">CY</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">CZ</font> =  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_6_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">kx</font>, <font color="Olive">ky</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x'</font>,<font color="Olive">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] holds <br/><font color="Olive">kx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">ky</font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_6_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_6_1_1_1_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">ky</font>, <font color="Olive">kz</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z'</font>,<font color="Olive">kz</font>] holds <br/><font color="Olive">ky</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">kz</font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_6_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_1_1_1_2"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">ky</font>, <font color="Olive">kz</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z'</font>,<font color="Olive">kz</font>] holds <br/><font color="Olive">ky</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">kz</font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_6_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:87_1_6_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_2"/><i><font color="Green">E85</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">x'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">kx</font>, <font color="Olive">ky</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x'</font>,<font color="Olive">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] holds <br/><font color="Olive">kx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">ky</font> ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_6_1_1_2_1_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">ky</font>, <font color="Olive">kz</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z'</font>,<font color="Olive">kz</font>] holds <br/><font color="Olive">ky</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">kz</font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_2_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:87_1_6_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_6_1_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">y'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">z'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">ky</font>, <font color="Olive">kz</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y'</font>,<font color="Olive">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z'</font>,<font color="Olive">kz</font>] holds <br/><font color="Olive">ky</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">kz</font> ) )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">zx</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">zy</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:87_1_6_1_1_2_1_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">zy</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">zz</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_6_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">kx</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">kz</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:87_1_6_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x'</font>,<font color="Maroon">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z'</font>,<font color="Maroon">kz</font>] )
 ;<br/><a NAME="E2:87_1_6_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">kx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">zy</font> &amp; <font color="Maroon">zy</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kz</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E3:87_1_6_1_1_2_1_1_2_1"/>
<font color="Maroon">kx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">kz</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:87_1_6_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:87_1_6_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:87_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 ;<br/></div><b>end;</b></div><a NAME="E12:87_1"/><b>then </b><i><font color="Green">E88</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 ;<br/><b>consider </b><font color="Maroon">M</font> being    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E14:87_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">M</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S4">CQC_SIM1:sch 4</a>(, , <a class="txt" href="glib_004.html#E7:87_1"><i><font color="Green">E66</font></i></a>);<br/></i><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E16:87_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="txt" href="glib_004.html#E8:87_1"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E18:87_1"/><i><font color="Green">E91</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_7"/><i><font color="Green">E93</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:87_1_7"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T89">GLIB_000:89</a></i>;<br/> <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G2</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
<br/>.= 
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">M</font>
<b>by </b><i>, <a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:87_1_7"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_8"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:87_1_8"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:87_1_8"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><a NAME="E4:87_1_8"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span>
 ;<br/><a NAME="E5:87_1_8"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">M</font> <a href="glib_000.html#K18">.size()</a> 
 ;<br/><a NAME="E6:87_1_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E7:87_1_8"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/><a NAME="E8:87_1_8"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:87_1_8"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E21:87_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:87_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_9"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:87_1_9"/><b>then </b><i><font color="Green">E98</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:87_1_9"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E4:87_1_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K12" target="_self">PRIM:Init</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>hence </b><a NAME="E5:87_1_9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">k1o</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:87_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1o</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k1o</font> = <font color="Maroon">k1o</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">Gk1b</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font>;<br/><b>set </b><font color="Maroon">Gk</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font>;<br/><a NAME="E28:87_1"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K13" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E29:87_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k1o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E30:87_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">Next</font> =  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>;<br/><b>set </b><font color="Maroon">ep</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>;<br/><a NAME="E31:87_1"/><i><font color="Green">E102</font></i>: 
 <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="glib_004.html#T2" target="_self">Th2</a>, , </i>;<br/><a NAME="E32:87_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span><b> such that </b><br/><a NAME="E34:87_1"/><i><font color="Green">E104</font></i>: 
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">v</font>,1 )
 <b>by </b><i>, <a class="txt" href="glib_004.html#E10:87_1"><i><font color="Green">E76</font></i></a>, </i>;<br/><i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
<b>by </b><i>, <a class="ref" href="glib_003.html#T56">GLIB_003:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>
;<br/><a NAME="E36:87_1"/><b>then </b><i><font color="Green">E106</font></i>: 
not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E37:87_1"/><b>then </b><i><font color="Green">E107</font></i>: 
not  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>set </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<br/><a NAME="E38:87_1"/><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E109</font></i>:  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> = 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</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">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Mep</font> being   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G1</font>, <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E42:87_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E43:87_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E44:87_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E45:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E46:87_1"/><b>then </b>
(  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/><a NAME="E47:87_1"/><b>then </b><i><font color="Green">E110</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Mep</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E48:87_1"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Mep</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as  <a href="glib_002.html#V1">connected</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">Mep</font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><a NAME="E50:87_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E51:87_1"/><b>then </b><i><font color="Green">E218</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E52:87_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Mep</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</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">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">M</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">M'</font><b> such that </b><br/><a NAME="E55:87_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>consider </b><font color="Maroon">PW</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W</font>;<br/><a NAME="E57:87_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">PW</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E58:87_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">PW</font> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 ;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">PW</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">Mep</font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E60:87_1"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:87_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/><a NAME="E61:87_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">v1</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E62:87_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E63:87_1"/><b>then </b><i><font color="Green">E227</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E64:87_1"/><i><font color="Green">E228</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" target="_self">PRIM:MST</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i/>;<br/><a NAME="E65:87_1"/><b>then </b><i><font color="Green">E229</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E18:87_1"><i><font color="Green">E91</font></i></a>, , <a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a></i>;<br/><a NAME="E66:87_1"/><i><font color="Green">E230</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">Mep</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E67:87_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><a NAME="E68:87_1"/><b>then </b><i><font color="Green">E231</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G2'</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E18:87_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="glib_004.html#E19:87_1"><i><font color="Green">E92</font></i></a>, , <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E69:87_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> ) )
 <b>by </b><i><a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E70:87_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E18:87_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="glib_004.html#E19:87_1"><i><font color="Green">E92</font></i></a>, , <a class="ref" href="glib_000.html#D20">GLIB_000:def 20</a></i>;<br/><a NAME="E71:87_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E72:87_1"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E73:87_1"/><b>then </b><i><font color="Green">E232</font></i>: 
not <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><a NAME="E74:87_1"/><i><font color="Green">E233</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">v2</font>,<font color="Maroon">Mep</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E75:87_1"/><i><font color="Green">E234</font></i>: 
not  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E14:87_1"><i><font color="Green">E89</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:87_1_12"/><i><font color="Green">E235</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 ;<br/><font color="Maroon">v2</font> = 
<font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>
;<br/><a NAME="E3:87_1_12"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>then </b><font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  = 
<font color="Maroon">v2</font>
<b>by </b><i><a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><a NAME="E5:87_1_12"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24">GLIB_001:def 24</a></i>;<br/><b>reconsider </b><font color="Maroon">PM</font> = <font color="Maroon">P</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">M</font> ;<br/><a NAME="E7:87_1_12"/>
( <font color="Maroon">PM</font> is <a href="glib_001.html#V5">Path-like</a> &amp; <font color="Maroon">PM</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">PM</font> is <a href="glib_001.html#V3">trivial</a> )
 <b>by </b><i><a class="txt" href="glib_004.html#E28:87_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#T127">GLIB_001:127</a>, <a class="ref" href="glib_001.html#T177">GLIB_001:177</a></i>;<br/><a NAME="E8:87_1_12"/><b>then </b>
<font color="Maroon">PM</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E9:87_1_12"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2">GLIB_002:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E77:87_1"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E21:87_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, , <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/><b>set </b><font color="Maroon">C</font> = <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E78:87_1"/><i><font color="Green">E240</font></i>: 
not <font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T133">GLIB_001:133</a></i>;<br/><a NAME="E79:87_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">v2</font>,<font color="Maroon">v2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E16:87_1"><i><font color="Green">E90</font></i></a>, , <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E80:87_1"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T120">GLIB_001:120</a></i>;<br/><a NAME="E81:87_1"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="glib_004.html#E28:87_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E82:87_1"/><i><font color="Green">E242</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R3">SJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:87_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="glib_004.html#E3:87"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E83:87_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/><a NAME="E84:87_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E85:87_1"/><b>then </b><i><font color="Green">E243</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E86:87_1"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_13"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_13_1"/>
( ( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> ) or ( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E31:87_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="glib_004.html#E32:87_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="glib_000.html#D17">GLIB_000:def 17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_13_1_1"><b>suppose </b></a><a NAME="E1:87_1_13_1_1"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  );<br/><a NAME="E2:87_1_13_1_1"/><i><font color="Green">E247</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E34:87_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a></i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:87_1_13_1_1"/><i><font color="Green">E249</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="glib_004.html#E36:87_1"><i><font color="Green">E106</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a></i>;<br/><a NAME="E6:87_1_13_1_1"/><i><font color="Green">E251</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E7:87_1_13_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="txt" href="glib_004.html#E34:87_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a></i>;<br/><a NAME="E8:87_1_13_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E38:87_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:87_1_13_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">m2k</font> = <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:87_1_13_1_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">m2k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E12:87_1_13_1_1"/><b>then </b><i><font color="Green">E254</font></i>: 
<font color="Maroon">m2k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:87_1_13_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_004.html#E39:87_1"><i><font color="Green">E109</font></i></a></i>;<br/><a NAME="E14:87_1_13_1_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><b>set </b><font color="Maroon">em</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">em</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E15:87_1_13_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E16:87_1_13_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:87_1_13_1_1"/>
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:87_1_13_1_1"/><i><font color="Green">E259</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font> )
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E20:87_1_13_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:87_1_13_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E22:87_1_13_1_1"/><i><font color="Green">E262</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E23:87_1_13_1_1"/><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E24:87_1_13_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E25:87_1_13_1_1"/><b>then </b><i><font color="Green">E266</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:87_1_13_1_1"/><i><font color="Green">E267</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E27:87_1_13_1_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V4">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E28:87_1_13_1_1"/>
<font color="Maroon">em</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, , , , , <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/><a NAME="E29:87_1_13_1_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">em</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font>,<font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E30:87_1_13_1_1"/><b>then </b>
<font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/><a NAME="E31:87_1_13_1_1"/><b>then </b>
<font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E32:87_1_13_1_1"/>
<font color="Maroon">em</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font>,<font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, , <a class="ref" href="glib_000.html#T20">GLIB_000:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_13_1_2"><b>suppose </b></a><a NAME="E1:87_1_13_1_2"/><i><font color="Green">E270</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> )
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> );<br/><a NAME="E2:87_1_13_1_2"/><i><font color="Green">E271</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E34:87_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a></i>;<br/><b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:87_1_13_1_2"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="glib_004.html#E36:87_1"><i><font color="Green">E106</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a></i>;<br/><a NAME="E6:87_1_13_1_2"/><i><font color="Green">E273</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E7:87_1_13_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E34:87_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:87_1_13_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E38:87_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:87_1_13_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">m2k</font> = <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:87_1_13_1_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">m2k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E12:87_1_13_1_2"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">m2k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green">E276</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_13_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_13_1_2_1"/><i><font color="Green">E277</font></i>: 
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:87_1_13_1_2_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E3:87_1_13_1_2_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:87_1_13_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_004.html#E39:87_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E14:87_1_13_1_2"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><b>set </b><font color="Maroon">em</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">em</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m2k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E15:87_1_13_1_2"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E16:87_1_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:87_1_13_1_2"/>
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:87_1_13_1_2"/><i><font color="Green">E280</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font> )
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E20:87_1_13_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:87_1_13_1_2"/><b>then </b><i><font color="Green">E281</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, , <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E22:87_1_13_1_2"/><i><font color="Green">E282</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E23:87_1_13_1_2"/><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:87_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E24:87_1_13_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E25:87_1_13_1_2"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:87_1_13_1_2"/><i><font color="Green">E292</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E27:87_1_13_1_2"/>
<font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V4">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:87_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E28:87_1_13_1_2"/>
<font color="Maroon">em</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, , , , , <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/><a NAME="E29:87_1_13_1_2"/>
<font color="Maroon">em</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m2k</font>,<font color="Maroon">PW</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><b>hence </b><a NAME="E30:87_1_13_1_2"/>
<font color="Maroon">em</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font>,<font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, , <a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_000.html#T20">GLIB_000:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">em</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E89:87_1"/><i><font color="Green">E293</font></i>: 
( <font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  &amp; <font color="Maroon">em</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon">em</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">V</font>,<font color="Maroon">M</font> )
 ;<br/><a NAME="E90:87_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">em</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font>,<font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75">GLIB_000:75</a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, </i>;<br/><a NAME="E91:87_1"/><b>then </b>
<font color="Maroon">em</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E92:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:87_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="glib_004.html#E3:87"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E93:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font>
 <b>by </b><i/>;<br/><a NAME="E94:87_1"/><b>then </b><i><font color="Green">E295</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">M2</font> being  <a href="glib_003.html#V1">[Weighted]</a> <a href="glib_003.html#V4">weight-inheriting</a> <a href="glib_000.html#NM8">removeEdge</a> of <font color="Maroon">Mep</font>,<font color="Maroon">em</font>;<br/><b>reconsider </b><font color="Maroon">M2</font> = <font color="Maroon">M2</font> as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_003.html#T16">GLIB_003:16</a></i>;<br/><a NAME="E97:87_1"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Mep</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as  <a href="glib_002.html#V1">connected</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">Mep</font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><a NAME="E99:87_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M'</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">Mep</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E100:87_1"/><b>then </b>
<font color="Maroon">M'</font> is <a href="glib_000.html#V9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E101:87_1"/><b>then </b>
<font color="Maroon">Mep</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T23">GLIB_002:23</a></i>;<br/><a NAME="E102:87_1"/><b>then </b><i><font color="Green">E296</font></i>: 
<font color="Maroon">M2</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E30:87_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="glib_002.html#T5">GLIB_002:5</a></i>;<br/><a NAME="E103:87_1"/><i><font color="Green">E297</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T54">GLIB_000:54</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_14"><b>now </b></a><div class="add"><a NAME="E1:87_1_14"/><i><font color="Green">E298</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><i><font color="Green">E299</font></i>: <font color="Maroon">M2</font> <a href="glib_000.html#K18">.size()</a>  = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_004.html#E39:87_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="glib_000.html#K18">.size()</a> 

;<br/><font color="Maroon">M2</font> <a href="glib_000.html#K16">.order()</a>  = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#T56">GLIB_000:56</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="glib_000.html#K16">.order()</a> 
<b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
;<br/><b>hence </b><a NAME="E4:87_1_14"/>
<font color="Maroon">M2</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="glib_000.html#K18">.size()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E105:87_1"/><b>then </b><i><font color="Green">E300</font></i>: 
<font color="Maroon">M2</font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E38:87_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><b>set </b><font color="Maroon">M2'</font> = <font color="Maroon">M2</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a> ;<br/><a NAME="E106:87_1"/><i><font color="Green">E301</font></i>: 
( <font color="Maroon">M2</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">M2</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" 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">M2</font> )
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E107:87_1"/><b>then </b><i><font color="Green">E302</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E39:87_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">M2'</font> = <font color="Maroon">M2</font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K8" target="_self">WGraphSelectors</a>  as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_003.html#T15">GLIB_003:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_15"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:87_1_15"/>
<font color="Maroon">M2'</font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E47:87_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_002.html#T48">GLIB_002:48</a></i>;<br/> <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M2'</font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M2</font>
<b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</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">G1</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#T56">GLIB_000:56</a></i>
;<br/><b>hence </b><a NAME="E3:87_1_15"/>
<font color="Maroon">M2'</font> is <a href="glib_000.html#V9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M2'</font> = <font color="Maroon">M2'</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">G1</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_16"><b>now </b></a><div class="add"><a NAME="E1:87_1_16"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/><a NAME="E2:87_1_16"/><b>then </b><i><font color="Green">E304</font></i>: 
(  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> &amp;  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> )
 ;<br/><a NAME="E3:87_1_16"/><i><font color="Green">E305</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E4:87_1_16"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E5:87_1_16"/><b>then </b>
<font color="Maroon">M</font> is   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">G1</font>, <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font>, <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:87_1_16"/><b>then </b><i><font color="Green">E307</font></i>: 
<font color="Maroon">Mep</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, , , <a class="txt" href="glib_004.html#E2:87"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E7:87_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Mep</font> <a href="glib_004.html#K10" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, </i>;<br/><a NAME="E8:87_1_16"/><b>then </b><i><font color="Green">E309</font></i>: 
<font color="Maroon">M2</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Mep</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span>
 ;<br/><a NAME="E9:87_1_16"/>
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">Mep</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><a NAME="E10:87_1_16"/><b>then </b><i><font color="Green">E310</font></i>: 
<font color="Maroon">M2</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E11:87_1_16"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E12:87_1_16"/>
<font color="Maroon">M2'</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div><a NAME="E112:87_1"/><b>then </b><i><font color="Green">E311</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M2'</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">em</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="glib_004.html#K10" target="_self">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G1</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:87_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G3</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">G1</font>;<br/><a NAME="E1:87_1_17"/>
<font color="Maroon">M</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G3</font> <a href="glib_004.html#K10" target="_self">.cost()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:87_1_17"/>
<font color="Maroon">M2'</font> <a href="glib_004.html#K10" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G3</font> <a href="glib_004.html#K10" target="_self">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M2'</font> = <font color="Maroon">M2'</font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">MG2'</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>;<br/><b>set </b><font color="Maroon">MG2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>;<br/><a NAME="E115:87_1"/><i><font color="Green">E312</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T50">XBOOLE_1:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_18"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_18"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_1_18"/><i><font color="Green">E313</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:87_1_18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:87_1_18"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:87_1_18"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><a NAME="E117:87_1"/><b>then </b><i><font color="Green">E314</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E118:87_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2'</font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K8" target="_self">WGraphSelectors</a> 
 <b>by </b><i/>;<br/><a NAME="E119:87_1"/><b>then </b>
<font color="Maroon">M2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_004.html#K9" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i/>;<br/><a NAME="E120:87_1"/><b>then </b><i><font color="Green">E315</font></i>: 
<font color="Maroon">M2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div><i><font color="Green">E316</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_19"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:87_1_19"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k1o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:87_1_19"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1o</font>
 ;<br/><a NAME="E3:87_1_19"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E5:87"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:87_1_19"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E317</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_20"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font>
 ;<br/><a NAME="E2:87_1_20"/><b>then </b><i><font color="Green">E319</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_1_20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:87_1_20_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">em</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:87_1_20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E4:87_1_20"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then </b><i><font color="Green">E320</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">Mep</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>

;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:87_1_20"/><i><font color="Green">E321</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">M2'</font>,<font color="Maroon">k2</font>]
 <b>by </b><i><a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, </i>;<br/><div><i><font color="Green">E322</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">Vr</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:87_1_20_3"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><a NAME="E2:87_1_20_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:87_1_20_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/><a NAME="E4:87_1_20_3"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">em</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:87_1_20_3"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E36:87_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="glib_000.html#D17">GLIB_000:def 17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_1_20_4"/><i><font color="Green">E425</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_20_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:87_1_20_4_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_20_4_1_1_1"><b>suppose </b></a><a NAME="E1:87_1_20_4_1_1_1"/><i><font color="Green">E426</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div class="add"><a NAME="E2:87_1_20_4_1_1_1"/><b>then </b><i><font color="Green">E427</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, </i>;<br/><a NAME="E3:87_1_20_4_1_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, , </i>;<br/><b>hence </b><a NAME="E4:87_1_20_4_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_20_4_1_1_2"><b>suppose </b></a><a NAME="E1:87_1_20_4_1_1_2"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:87_1_20_4_1_1_2"/><b>then </b><i><font color="Green">E429</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:87_1_20_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:87_1_20_4_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E35:87_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:87_1_20_4_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:87_1_20_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font>
 ;<br/></div><b>end;</b></div><a NAME="E10:87_1_20"/><b>then </b><i><font color="Green">E430</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green">E431</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_20_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_20_5"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:87_1_20_5"/><b>then </b>
<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:87_1_20_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E5:87"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:87_1_20_5"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_004.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E12:87_1_20"/><i><font color="Green">E432</font></i>: 
<font color="Maroon">k1o</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, , , <a class="ref" href="glib_004.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:87_1_20"/>
<span class="p1">(<span class="default"><font color="Maroon">k1o</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E14:87_1_20"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_21"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_21"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_1_21"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:87_1_21"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">em</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:87_1_21"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E124:87_1"/><b>then </b><i><font color="Green">E434</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_22"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_22"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_1_22"/><i><font color="Green">E435</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:87_1_22"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:87_1_22"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:87_1_22"/>
contradiction
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><a NAME="E126:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M2'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/><a NAME="E128:87_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_004.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E129:87_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:87"/>
<font color="Maroon">G2</font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">G1</font>
 ;<br/>


</div>
