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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> =  <a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<i><font color="Green">E56</font></i>:  <a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K41">.Result()</a> 

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>

;<br/>
<a NAME="E2:84"/><b>then </b>
(  <a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/>
<a NAME="E3:84"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></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#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E68"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E5:84"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<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#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E3:84"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E6:84"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="glib_004.html#E4:84"><i><font color="Green">E58</font></i></a>, <a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/>
<a NAME="E7:84"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="glib_004.html#E68"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E8:84"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  )
 
<b>by </b><i><a class="txt" href="glib_004.html#E4:84"><i><font color="Green">E58</font></i></a>, <a class="txt" href="glib_004.html#E6:84"><i><font color="Green">E59</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E9:84"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_002.html#V3">Tree-like</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="glib_000.html#V9">spanning</a> )
 
<b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_004.html#E67"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="glib_002.html#V3">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_004.html#E67"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is  not   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:84_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:84_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a>  : <font color="Olive">b<sub>1</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> </span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> ;<br/><a NAME="E2:84_2_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E37"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_004.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E4:84_2_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E38"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E5:84_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E6:84_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>hence </b><a NAME="E7:84_2_2"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> </span>)</span> ;<br/><b>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="nat_1.html#NM1">Nat</a>] means ( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Olive">b<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#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>] means (  <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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) );<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E1:84_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:84_2_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_3"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:84_2_3"/><i><font color="Green">E65</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_3_1"/><i><font color="Green">E67</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:84_2_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="glib_004.html#E5:84_2_3"><i><font color="Green">E65</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">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:84_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="glib_004.html#E2:84_2_3_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="glib_004.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">c<sub>11</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_3_2"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:84_2_3_2"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:84_2_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">c<sub>3</sub></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">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><a NAME="E4:84_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></font></span>)</span>
 ;<br/><a NAME="E5:84_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K18">.size()</a> 
 ;<br/><a NAME="E6:84_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></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="E7:84_2_3_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></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#E9:84"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/><font color="Maroon">c<sub>3</sub></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#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E5:84_2_3"><i><font color="Green">E65</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K16">.order()</a> 

;<br/><b>hence </b><a NAME="E9:84_2_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E7:84_2_3_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E8:84_2_3"/><b>then </b><i><font color="Green">E67</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:84_2_3"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>12</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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#E8:84_2_3"><i><font color="Green">E67</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_3_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:84_2_3_3"/><b>then </b><i><font color="Green">E69</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:84_2_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="glib_004.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E3:84_2_3_3"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E54"><i><font color="Green">Lemma38</font></i></a></i>;<br/><a NAME="E4:84_2_3_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></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="E5:84_2_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E2:84_2_3_3"><i><font color="Green">E69</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">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:84_2_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E14:84_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_004.html#E13:84_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E15:84_2_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E10:84_2_3"><i><font color="Green">E68</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_2_3_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:84_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>14</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E63"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_3_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>14</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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E15:84_2_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:84_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E10:84_2_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E13:84_2_3"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:84_2_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:84_2_4"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_4"><i><font color="Green">E65</font></i></a></i>;<br/><b>assume </b><a NAME="E5:84_2_4"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>] )
 ;<br/><a NAME="E6:84_2_4"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E7:84_2_4"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:84_2_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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="E8:84_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E1:84_2_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:84_2_5"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E4:84_2_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:84_2_5"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></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">c<sub>11</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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></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">c<sub>12</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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_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">c<sub>11</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></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">c<sub>11</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">c<sub>3</sub></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">c<sub>11</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></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="84_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_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">c<sub>11</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_5_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E6:84_2_5"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_5_1_1_2"/><i><font color="Green">E68</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">c<sub>12</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">c<sub>3</sub></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">c<sub>11</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:84_2_5_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>15</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:84_2_5_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2"><i><font color="Green">E63</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_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="E1:84_2_5_1_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_5_1_1_2_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_2_5_1_1_2_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>] )
 ;<br/><a NAME="E2:84_2_5_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_5_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_004.html#E5:84_2_5_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2"><i><font color="Green">E64</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_5_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_5_1_1_2_1_1_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:84_2_5_1_1_2_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_5_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E3:84_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E6:84_2_5"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_5_1_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_5_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_2_5_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>17</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>18</sub></font>] )
 ;<br/><a NAME="E2:84_2_5_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_5_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_004.html#E5:84_2_5_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2"><i><font color="Green">E64</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_5_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_5_1_1_2_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:84_2_5_1_1_2_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_5_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E3:84_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E6:84_2_5"><i><font color="Green">E67</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="E7:84_2_5_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_5_1_1_3"><b>suppose </b></a><a NAME="E1:84_2_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">c<sub>11</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_5_1_1_3"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E6:84_2_5"><i><font color="Green">E67</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="E8:84_2_5"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 ;<br/></div><b>end;</b></div><a NAME="E10:84_2"/><b>then </b><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font> holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>assume </b><a NAME="E1:84_2_6"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>] )
 ;<br/><a NAME="E2:84_2_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:84_2_6"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E5:84_2_6"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:84_2_6"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E8:84_2_6"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:84_2_6"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></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">c<sub>12</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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></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">c<sub>13</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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_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">c<sub>12</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></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">c<sub>12</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_6"><i><font color="Green">E67</font></i></a>, <a class="txt" href="glib_004.html#E4:84_2_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2_6"><i><font color="Green">E69</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_6_1_1_1"/><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">c<sub>12</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_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="E1:84_2_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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_6"><i><font color="Green">E67</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2_6"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_6_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_6_1_1_1"><i><font color="Green">E71</font></i></a>, <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="84_2_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_6_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_6_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="glib_004.html#E4:84_2_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</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="E3:84_2_6_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_6_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">c<sub>12</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_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="E1:84_2_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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_6"><i><font color="Green">E67</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2_6"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_6_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_6_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_6_1_1_2_1_1_2"/><i><font color="Green">E72</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">c<sub>13</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">c<sub>3</sub></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">c<sub>14</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">c<sub>3</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:84_2_6_1_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>19</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2"><i><font color="Green">E63</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_6_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_2_6_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>21</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font>] )
 ;<br/><a NAME="E2:84_2_6_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E7:84_2_6"><i><font color="Green">E69</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_6_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_6_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_004.html#E4:84_2_6_1_1_2_1_1_2"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E3:84_2_6_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>22</sub></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="E7:84_2_6_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_6_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_6_1_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="glib_004.html#E4:84_2_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_6"><i><font color="Green">E70</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="E3:84_2_6_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:84_2_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 ;<br/></div><b>end;</b></div><a NAME="E12:84_2"/><b>then </b><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E14:84_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="glib_004.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Olive">b<sub>1</sub></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#E8:84_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="glib_004.html#E12:84_2"><i><font color="Green">E67</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E16:84_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E14:84_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E16:84_2"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E18:84_2"/><i><font color="Green">E70</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_7"/><i><font color="Green">E72</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:84_2_7"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E18:84_2"><i><font color="Green">E70</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">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_7"><i><font color="Green">E72</font></i></a>, <a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:84_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="glib_004.html#E2:84_2_7"><i><font color="Green">E73</font></i></a>, <a class="ref" href="glib_004.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</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">c<sub>11</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_8"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:84_2_8"/><b>then </b>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R2">c&lt;</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:84_2_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">c<sub>3</sub></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">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><a NAME="E4:84_2_8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></font></span>)</span>
 ;<br/><a NAME="E5:84_2_8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="glib_000.html#K18">.size()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K18">.size()</a> 
 ;<br/><a NAME="E6:84_2_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></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="E7:84_2_8"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>11</sub></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#E9:84"><i><font color="Green">E61</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/><font color="Maroon">c<sub>3</sub></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#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E18:84_2"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K16">.order()</a> 

;<br/><b>hence </b><a NAME="E9:84_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E7:84_2_8"><i><font color="Green">E72</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E21:84_2"/><b>then </b><i><font color="Green">E72</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E23:84_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>12</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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#E21:84_2"><i><font color="Green">E72</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_9"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:84_2_9"/><b>then </b><i><font color="Green">E74</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:84_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="glib_004.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E3:84_2_9"/>
 <a href="glib_003.html#K6">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E54"><i><font color="Green">Lemma38</font></i></a></i>;<br/><a NAME="E4:84_2_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K11" target="_self">PRIM:Init</a> <font color="Maroon">c<sub>1</sub></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="E5:84_2_9"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E2:84_2_9"><i><font color="Green">E74</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">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:84_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></font>;<br/><a NAME="E27:84_2"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K12" target="_self">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E26:84_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="glib_004.html#D15" target="_self">Def15</a></i>;<br/><a NAME="E28:84_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_004.html#E26:84_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E29:84_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:84_2"><i><font color="Green">E73</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>;<br/><a NAME="E30:84_2"/><i><font color="Green">E77</font></i>: 
 <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="txt" href="glib_004.html#E27:84_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self">Def14</a>, <a class="txt" href="glib_004.html#E29:84_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="glib_004.html#E23:84_2"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E31:84_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span><b> such that </b><br/><a NAME="E33:84_2"/><i><font color="Green">E79</font></i>: 
( not <font color="Maroon">c<sub>18</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <font color="Maroon">c<sub>18</sub></font>,1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E27:84_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="glib_004.html#E30:84_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="glib_004.html#E56"><i><font color="Green">Lemma40</font></i></a></i>;<br/><i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E33:84_2"><i><font color="Green">E79</font></i></a>, <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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E31:84_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_003.html#T52">GLIB_003:52</a></i>
;<br/><a NAME="E35:84_2"/><b>then </b><i><font color="Green">E81</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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:84_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="glib_004.html#E29:84_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E36:84_2"/><b>then </b><i><font color="Green">E82</font></i>: 
not  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>11</sub></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">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<br/><a NAME="E37:84_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><b>then </b><i><font color="Green">E84</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green">Lemma43</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>21</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>21</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E41:84_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E31:84_2"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E42:84_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:84_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E43:84_2"/><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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E44:84_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E45:84_2"/><b>then </b>
(  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/><a NAME="E46:84_2"/><b>then </b><i><font color="Green">E85</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>21</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E47:84_2"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="glib_002.html#V1">connected</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>21</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><a NAME="E49:84_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E50:84_2"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E51:84_2"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>21</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>21</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E50:84_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>24</sub></font><b> such that </b><br/><a NAME="E54:84_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></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">c<sub>28</sub></font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">c<sub>27</sub></font>;<br/><a NAME="E56:84_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E54:84_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><a NAME="E57:84_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>28</sub></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">c<sub>11</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">c<sub>21</sub></font> <b>by </b><i><a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E59:84_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>29</sub></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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E57:84_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="glib_001.html#T111">GLIB_001:111</a></i>;<br/><a NAME="E60:84_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E56:84_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E61:84_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E62:84_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E34:84_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E63:84_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></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#K14" target="_self">PRIM:MST</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T52" target="_self">Th52</a></i>;<br/><a NAME="E64:84_2"/><b>then </b><i><font color="Green">E94</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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E62:84_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T52">ZFMISC_1:52</a></i>;<br/><a NAME="E65:84_2"/><i><font color="Green">E95</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E50:84_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E66:84_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/><a NAME="E67:84_2"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E62:84_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="glib_004.html#E63:84_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a>, <a class="ref" href="glib_000.html#T76">GLIB_000:76</a></i>;<br/><a NAME="E68:84_2"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> ) )
 <b>by </b><i><a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E69:84_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E8:84"><i><font color="Green">E60</font></i></a>, <a class="txt" href="glib_004.html#E62:84_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="glib_004.html#E63:84_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="glib_004.html#E67:84_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="glib_000.html#D20">GLIB_000:def 20</a></i>;<br/><a NAME="E70:84_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E71:84_2"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E72:84_2"/><b>then </b><i><font color="Green">E97</font></i>: 
not <font color="Maroon">c<sub>29</sub></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="E73:84_2"/><i><font color="Green">E98</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K4">.last()</a> ,<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="glib_004.html#E65:84_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E74:84_2"/><i><font color="Green">E99</font></i>: 
not  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E59:84_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="glib_004.html#E35:84_2"><i><font color="Green">E81</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="84_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_2_12"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 ;<br/><font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></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="E3:84_2_12"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_12"><i><font color="Green">E100</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>then </b><font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K4">.last()</a>  = 
<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_12"><i><font color="Green">E100</font></i></a>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><a NAME="E5:84_2_12"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>29</sub></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">c<sub>31</sub></font> = <font color="Maroon">c<sub>29</sub></font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">c<sub>11</sub></font> ;<br/><a NAME="E7:84_2_12"/>
( <font color="Maroon">c<sub>31</sub></font> is <a href="glib_001.html#V5">Path-like</a> &amp; <font color="Maroon">c<sub>31</sub></font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">c<sub>31</sub></font> is <a href="glib_001.html#V3">trivial</a> )
 <b>by </b><i><a class="txt" href="glib_004.html#E5:84_2_12"><i><font color="Green">E101</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_12"><i><font color="Green">E100</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="E8:84_2_12"/><b>then </b>
<font color="Maroon">c<sub>31</sub></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="E9:84_2_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="E76:84_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E72:84_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="glib_004.html#E74:84_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="glib_001.html#T151">GLIB_001:151</a></i>;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E77:84_2"/><i><font color="Green">E101</font></i>: 
not <font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T133">GLIB_001:133</a></i>;<br/><a NAME="E78:84_2"/>
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E60:84_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="glib_004.html#E65:84_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="glib_001.html#T67">GLIB_001:67</a></i>;<br/><a NAME="E79:84_2"/><b>then </b>
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E80:84_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E76:84_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="glib_004.html#E77:84_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E81:84_2"/><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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E30:84_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E82:84_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E83:84_2"/><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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E84:84_2"/><b>then </b><i><font color="Green">E104</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E50:84_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><a NAME="E85:84_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></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">c<sub>29</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E56:84_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_13"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_13_1"/>
( ( <font color="Maroon">c<sub>25</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">c<sub>26</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> ) or ( <font color="Maroon">c<sub>26</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">c<sub>25</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E81:84_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="glib_004.html#E84:84_2"><i><font color="Green">E104</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="84_2_13_1_1"><b>suppose </b></a><a NAME="E1:84_2_13_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>25</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">c<sub>26</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="nat_1.html#NM1">Nat</a>] means ( 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">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  );<br/><a NAME="E2:84_2_13_1_1"/><i><font color="Green">E107</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E85:84_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_13_1_1"><i><font color="Green">E106</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:84_2_13_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>31</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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#E2:84_2_13_1_1"><i><font color="Green">E107</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E6:84_2_13_1_1"/><i><font color="Green">E109</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E7:84_2_13_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="txt" href="glib_004.html#E85:84_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_13_1_1"><i><font color="Green">E106</font></i></a></i>;<br/><a NAME="E8:84_2_13_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2_13_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:84_2_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">c<sub>32</sub></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">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></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="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:84_2_13_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></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="E12:84_2_13_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:84_2_13_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
not <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_004.html#E11:84_2_13_1_1"><i><font color="Green">E110</font></i></a></i>;<br/><a NAME="E14:84_2_13_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E15:84_2_13_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_13_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E16:84_2_13_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:84_2_13_1_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E15:84_2_13_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>36</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:84_2_13_1_1"/><i><font color="Green">E115</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E15:84_2_13_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E20:84_2_13_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:84_2_13_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="glib_004.html#E19:84_2_13_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E22:84_2_13_1_1"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E23:84_2_13_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E24:84_2_13_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></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">c<sub>29</sub></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="E25:84_2_13_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>36</sub></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">c<sub>29</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:84_2_13_1_1"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></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">c<sub>29</sub></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="E27:84_2_13_1_1"/>
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E76:84_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E28:84_2_13_1_1"/>
<font color="Maroon">c<sub>35</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_004.html#E22:84_2_13_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E21:84_2_13_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_004.html#E23:84_2_13_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_004.html#E25:84_2_13_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_004.html#E26:84_2_13_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/><a NAME="E29:84_2_13_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_13_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_004.html#E14:84_2_13_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E30:84_2_13_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/><a NAME="E31:84_2_13_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></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">c<sub>11</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E13:84_2_13_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E32:84_2_13_1_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>20</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_004.html#E29:84_2_13_1_1"><i><font color="Green">E121</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="84_2_13_1_2"><b>suppose </b></a><a NAME="E1:84_2_13_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>26</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">c<sub>25</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="nat_1.html#NM1">Nat</a>] means ( 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">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> );<br/><a NAME="E2:84_2_13_1_2"/><i><font color="Green">E107</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E85:84_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_13_1_2"><i><font color="Green">E106</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:84_2_13_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>31</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></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#E2:84_2_13_1_2"><i><font color="Green">E107</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a></i>;<br/><a NAME="E6:84_2_13_1_2"/><i><font color="Green">E109</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E7:84_2_13_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E85:84_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_13_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:84_2_13_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2_13_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:84_2_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">c<sub>32</sub></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">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></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="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:84_2_13_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></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="E12:84_2_13_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_13_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_13_1_2_1"/><i><font color="Green">E113</font></i>: 
not <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><a NAME="E2:84_2_13_1_2_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_13_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_004.html#E38:84_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E3:84_2_13_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_13_1_2_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:84_2_13_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_004.html#E11:84_2_13_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_004.html#E12:84_2_13_1_2"><i><font color="Green">E111</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E14:84_2_13_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E15:84_2_13_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_13_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E16:84_2_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:84_2_13_1_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E15:84_2_13_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>36</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:84_2_13_1_2"/><i><font color="Green">E115</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E15:84_2_13_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_001.html#T100">GLIB_001:100</a></i>;<br/><a NAME="E20:84_2_13_1_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:84_2_13_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="glib_004.html#E19:84_2_13_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E22:84_2_13_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T65">GLIB_001:65</a></i>;<br/><a NAME="E23:84_2_13_1_2"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E73:84_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="glib_001.html#T66">GLIB_001:66</a></i>;<br/><a NAME="E24:84_2_13_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></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">c<sub>29</sub></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="E25:84_2_13_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>36</sub></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">c<sub>29</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E26:84_2_13_1_2"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>29</sub></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">c<sub>29</sub></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="E27:84_2_13_1_2"/>
<font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E76:84_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E28:84_2_13_1_2"/>
<font color="Maroon">c<sub>35</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E19:84_2_13_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_004.html#E22:84_2_13_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E21:84_2_13_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_004.html#E23:84_2_13_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_004.html#E25:84_2_13_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_004.html#E26:84_2_13_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/><a NAME="E29:84_2_13_1_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_13_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_004.html#E14:84_2_13_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><b>hence </b><a NAME="E30:84_2_13_1_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>20</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E4:84_2_13_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_004.html#E38:84_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="glib_004.html#E13:84_2_13_1_2"><i><font color="Green">E112</font></i></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">c<sub>31</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E88:84_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14">.edges()</a>  &amp; <font color="Maroon">c<sub>31</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>20</sub></font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 ;<br/><a NAME="E89:84_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>20</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75">GLIB_000:75</a>, <a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E38:84_2"><i><font color="Green">E84</font></i></a></i>;<br/><a NAME="E90:84_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">c<sub>20</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>20</sub></font>,<span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:84_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E91:84_2"/><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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E30:84_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_004.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E92:84_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E93:84_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green">Lemma42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>32</sub></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">c<sub>21</sub></font>,<font color="Maroon">c<sub>31</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_003.html#T16">GLIB_003:16</a></i>;<br/><a NAME="E96:84_2"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>21</sub></font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="glib_002.html#V1">connected</a>  <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>21</sub></font> <b>by </b><i><a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/><a NAME="E98:84_2"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E99:84_2"/><b>then </b>
<font color="Maroon">c<sub>34</sub></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="E100:84_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></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="E101:84_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>33</sub></font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E80:84_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="glib_002.html#T5">GLIB_002:5</a></i>;<br/><a NAME="E102:84_2"/><i><font color="Green">E110</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>33</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_000.html#T54">GLIB_000:54</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_14"><b>now </b></a><div class="add"><a NAME="E1:84_2_14"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><i><font color="Green">E112</font></i>: <font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K18">.size()</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">c<sub>33</sub></font></span>)</span>

<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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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> <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">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_14"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_004.html#E102:84_2"><i><font color="Green">E110</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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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_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">c<sub>11</sub></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="txt" href="glib_004.html#E36:84_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K18">.size()</a> 

;<br/><font color="Maroon">c<sub>33</sub></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#K25">the_Vertices_of</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>

<br/>.= 
 <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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_000.html#T56">GLIB_000:56</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_000.html#K16">.order()</a> 

;<br/><b>hence </b><a NAME="E4:84_2_14"/>
<font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></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#E2:84_2_14"><i><font color="Green">E112</font></i></a>, <a class="ref" href="glib_002.html#T46">GLIB_002:46</a></i>;<br/></div><b>end;</b></div><a NAME="E104:84_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>33</sub></font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E101:84_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="glib_002.html#T47">GLIB_002:47</a></i>;<br/><b>set </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> ;<br/><a NAME="E105:84_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">c<sub>33</sub></font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E37"><i><font color="Green">Lemma30</font></i></a></i>;<br/><a NAME="E106:84_2"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1">=</a> <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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E102:84_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>33</sub></font> <a href="glib_000.html#K14">.strict</a> <a href="glib_004.html#K7" target="_self">WGraphSelectors</a>  as   <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="glib_004.html#E105:84_2"><i><font color="Green">E112</font></i></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="84_2_15"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:84_2_15"/>
<font color="Maroon">c<sub>36</sub></font> is <a href="glib_002.html#V3">Tree-like</a>
 <b>by </b><i><a class="txt" href="glib_004.html#E104:84_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_004.html#E105:84_2"><i><font color="Green">E112</font></i></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">c<sub>36</sub></font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>33</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E105:84_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="glib_000.html#T56">GLIB_000:56</a></i>
;<br/><b>hence </b><a NAME="E3:84_2_15"/>
<font color="Maroon">c<sub>36</sub></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">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font> as  <a href="glib_000.html#V9">spanning</a> <a href="glib_002.html#V3">Tree-like</a> <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_16"><b>now </b></a><div class="add"><a NAME="E1:84_2_16"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/><a NAME="E2:84_2_16"/><b>then </b><i><font color="Green">E115</font></i>: 
(  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> &amp;  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E3:84_2_16"/><i><font color="Green">E116</font></i>: 
 <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E37:84_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E4:84_2_16"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/><a NAME="E5:84_2_16"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is   <a href="glib_003.html#NM15">inducedWSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font>, <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_16"><i><font color="Green">E114</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E6:84_2_16"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_004.html#K9" 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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_16"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_004.html#E31:84_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_16"><i><font color="Green">E114</font></i></a>, <a class="txt" href="glib_004.html#E2:84_2_16"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_004.html#E36:84_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="glib_004.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E7:84_2_16"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="glib_004.html#K9" 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">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="glib_004.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E8:84_2_16"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="glib_004.html#K9" 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">c<sub>21</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 ;<br/><a NAME="E9:84_2_16"/>
 <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">c<sub>21</sub></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">c<sub>1</sub></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">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10">GLIB_003:def 10</a></i>;<br/><a NAME="E10:84_2_16"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="glib_004.html#K9" 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">c<sub>11</sub></font> <a href="glib_004.html#K9" 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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E6:84_2_16"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E8:84_2_16"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E11:84_2_16"/>
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E105:84_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E12:84_2_16"/>
<font color="Maroon">c<sub>37</sub></font> <a href="glib_004.html#K9" 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">c<sub>11</sub></font> <a href="glib_004.html#K9" 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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E105:84_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_16"><i><font color="Green">E119</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E111:84_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="glib_004.html#K9" 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">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></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">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></font> <a href="glib_004.html#K9" 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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E93:84_2"><i><font color="Green">E108</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="84_2_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be  <a href="glib_000.html#V9">spanning</a> <a href="glib_002.html#V3">Tree-like</a> <a href="glib_003.html#NM8">WSubgraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:84_2_17"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/><b>hence </b><a NAME="E2:84_2_17"/>
<font color="Maroon">c<sub>37</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> <a href="glib_004.html#K9" target="_self">.cost()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E111:84_2"><i><font color="Green">E114</font></i></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">c<sub>38</sub></font> = <font color="Maroon">c<sub>37</sub></font> as   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="glib_004.html#D17" target="_self">Def17</a></i>;<br/><b>set </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></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">c<sub>3</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>40</sub></font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E114:84_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></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">c<sub>3</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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">c<sub>31</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E106:84_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xboole_1.html#T50">XBOOLE_1:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_18"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_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">c<sub>11</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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>41</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_18"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>41</sub></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">c<sub>11</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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E4:84_2_18"/>
( <font color="Maroon">c<sub>41</sub></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">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>41</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E3:84_2_18"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:84_2_18"/><b>then </b>
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>41</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E6:84_2_18"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E35:84_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><a NAME="E116:84_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E117:84_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_004.html#K7" target="_self">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E38"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E118:84_2"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_004.html#K8" target="_self">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E119:84_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div><i><font color="Green">E118</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_19"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:84_2_19"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</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#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E23:84_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="glib_004.html#E26:84_2"><i><font color="Green">E74</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>41</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E2:84_2_19"/>
<font color="Maroon">c<sub>41</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E3:84_2_19"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>41</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E63"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:84_2_19"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>41</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">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E29:84_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_20"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_20"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:84_2_20"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></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">c<sub>3</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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">c<sub>31</sub></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>, <a class="txt" href="glib_004.html#E114:84_2"><i><font color="Green">E115</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>41</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_20_1"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:84_2_20_1"/><b>then </b>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:84_2_20_1"/>
<font color="Maroon">c<sub>41</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_20"><i><font color="Green">E120</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="E4:84_2_20"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></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">E122</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">c<sub>38</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">c<sub>3</sub></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">c<sub>21</sub></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">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="glib_004.html#E2:84_2_20"><i><font color="Green">E121</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">c<sub>21</sub></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">c<sub>3</sub></font></span>)</span></span>)</span></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/>.= 
<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">c<sub>11</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="txt" href="glib_004.html#E64:84_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="glib_004.html#E46:84_2"><i><font color="Green">E85</font></i></a>, <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">c<sub>11</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="txt" href="glib_004.html#E116:84_2"><i><font color="Green">E116</font></i></a>, <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">c<sub>11</sub></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">c<sub>3</sub></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">c<sub>11</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">c<sub>3</sub></font></span>)</span></span>)</span>

;<br/><b>consider </b><font color="Maroon">c<sub>41</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:84_2_20"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>38</sub></font>,<font color="Maroon">c<sub>41</sub></font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E119:84_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E6:84_2"><i><font color="Green">E63</font></i></a></i>;<br/><div><i><font color="Green">E124</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_20_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>42</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>20</sub></font>;<br/><b>assume </b><a NAME="E1:84_2_20_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>31</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><a NAME="E2:84_2_20_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E59"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E3:84_2_20_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_20_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="glib_000.html#T34">GLIB_000:34</a></i>;<br/><a NAME="E4:84_2_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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>20</sub></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="E5:84_2_20_3"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E89:84_2"><i><font color="Green">E107</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="84_2_20_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>42</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_20_4"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>42</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_20_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_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="E1:84_2_20_4_1_1"/>
( <font color="Maroon">c<sub>42</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  or <font color="Maroon">c<sub>42</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_20_4"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="txt" href="glib_004.html#E34:84_2"><i><font color="Green">E80</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_20_4_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_20_4_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>42</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#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 ;<br/><div class="add"><a NAME="E2:84_2_20_4_1_1_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>42</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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>, <a class="txt" href="glib_004.html#E29:84_2"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E3:84_2_20_4_1_1_1"/>
not <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="glib_004.html#E8:84_2_20"><i><font color="Green">E124</font></i></a>, <a class="txt" href="glib_004.html#E1:84_2_20_4_1_1_1"><i><font color="Green">E126</font></i></a></i>;<br/><b>hence </b><a NAME="E4:84_2_20_4_1_1_1"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:84_2_20_4_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_004.html#E106:84_2"><i><font color="Green">E113</font></i></a>, <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="84_2_20_4_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_20_4_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>42</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:84_2_20_4_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>42</sub></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">c<sub>11</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E3:84_2_20_4_1_1_2"/>
<font color="Maroon">c<sub>42</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:84_2_20_4_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:84_2_20_4_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_004.html#E88:84_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:84_2_20_4_1_1_2"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E106:84_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="glib_004.html#E2:84_2_20_4_1_1_2"><i><font color="Green">E127</font></i></a>, <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="E3:84_2_20_4"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E10:84_2_20"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</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">c<sub>38</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green">E126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_20_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_20_5"/>
<font color="Maroon">c<sub>41</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:84_2_20_5"/><b>then </b>
<font color="Maroon">c<sub>41</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:84_2_20_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>41</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E63"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:84_2_20_5"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E7:84_2_20"><i><font color="Green">E123</font></i></a>, <a class="txt" href="glib_004.html#E10:84_2_20"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E12:84_2_20"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="glib_004.html#E5:84_2_20"><i><font color="Green">E122</font></i></a>, <a class="txt" href="glib_004.html#E119:84_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E120:84_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_004.html#E14:84_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="glib_004.html#E7:84_2_20"><i><font color="Green">E123</font></i></a></i>;<br/><a NAME="E13:84_2_20"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="txt" href="glib_004.html#E26:84_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E14:84_2_20"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E12:84_2_20"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_004.html#E11:84_2_20"><i><font color="Green">E126</font></i></a>, <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="84_2_21"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_21"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>3</sub></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">c<sub>41</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_21"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></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">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:84_2_21"/>
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E3:84_2_21"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:84_2_21"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E121:84_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E123:84_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>38</sub></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">c<sub>3</sub></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">c<sub>11</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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_004.html#E114:84_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_22"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_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">c<sub>11</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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>41</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_22"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>41</sub></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">c<sub>11</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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E4:84_2_22"/>
( <font color="Maroon">c<sub>41</sub></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">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>41</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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#E3:84_2_22"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:84_2_22"/><b>then </b>
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>41</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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="E6:84_2_22"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E35:84_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div><a NAME="E125:84_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></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">c<sub>38</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">c<sub>3</sub></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">c<sub>11</sub></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">c<sub>3</sub></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#K10" target="_self">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K13" target="_self">PRIM:CompSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="glib_000.html#K42">.-&gt;</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_004.html#E123:84_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_004.html#E64:84_2"><i><font color="Green">E94</font></i></a>, <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">c<sub>11</sub></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">c<sub>3</sub></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="E127:84_2"/><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">c<sub>11</sub></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">c<sub>3</sub></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">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_004.html#E119:84_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="glib_004.html#E14:84_2"><i><font color="Green">E68</font></i></a></i>;<br/><b>hence </b><a NAME="E128:84_2"/>
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="E12:84"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="glib_004.html#NM2">minimumSpanningTree</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
