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

<b>let </b><font color="Maroon" title="c1">G1</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being  <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  holds  <font color="Olive" title="b1">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font></span><br/>

<b>set </b><font color="Maroon" title="c2">PMST</font> =  <a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c3">VG1</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c4">EG1</font> =  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">WG1</font> =  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>let </b><font color="Maroon" title="c6">G2</font> be   <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font></span><br/>

<b>set </b><font color="Maroon" title="c7">PCS</font> =  <a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font>;<br/>
<a NAME="E1:83"/><i><font color="Green" title="E56">A2</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#T39" target="_self" title="GLIB_004:th.39">Lm18</a></i>;<br/>
<a NAME="E2:83"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E3:83"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font>
 
;<br/>
<a NAME="E4:83"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="glib_000.html#T37" title="GLIB_000:th.37">GLIB_000:37</a></i>;<br/>
<a NAME="E5:83"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="ref" href="glib_004.html#T39" target="_self" title="GLIB_004:th.39">Lm18</a></i>;<br/>
<a NAME="E6:83"/><b>then </b><i><font color="Green" title="E58">A4</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 
<b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="#E4:83"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E7:83"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
( <font color="Maroon" title="c6">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> &amp; <font color="Maroon" title="c6">G2</font> is <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> )
 
<b>by </b><i><a class="ref" href="glib_004.html#T38" target="_self" title="GLIB_004:th.38">Lm17</a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G2'</font> = <font color="Maroon" title="c6">G2</font> as  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <b>by </b><i><a class="ref" href="glib_004.html#T38" target="_self" title="GLIB_004:th.38">Lm17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c6">G2</font> is  not   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>set </b><font color="Maroon" title="c9">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">x</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">x</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> </span>}</span>  implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  )</span><br/><b>assume </b><a NAME="E1:83_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">x</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span><br/><b>then consider </b><font color="Maroon" title="c11">G2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:83_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G2</font> &amp; <font color="Maroon" title="c11">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> )
 ;<br/><b>thus </b><a NAME="E4:83_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 <b>by </b><i><a class="txt" href="#E3:83_1_1"><i><font color="Green" title="E61">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  : <font color="Olive" title="b1">x</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c11">M</font> being   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font>;<br/><b>set </b><font color="Maroon" title="c12">M'</font> = <font color="Maroon" title="c11">M</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ;<br/><a NAME="E2:83_1_2"/>
( <font color="Maroon" title="c11">M</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c11">M</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">M</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c11">M</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E17"><i><font color="Green" title="E13">Lm2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">M'</font> = <font color="Maroon" title="c11">M</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  as   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="ref" href="glib_004.html#T41" target="_self" title="GLIB_004:th.41">Th36</a></i>;<br/><a NAME="E4:83_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E18"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/><a NAME="E5:83_1_2"/><b>then </b>
<font color="Maroon" title="c13">M'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E6:83_1_2"/><b>then </b>
<font color="Maroon" title="c13">M'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">X</font>
 ;<br/><b>hence </b><a NAME="E7:83_1_2"/>
<font color="Maroon" title="c10">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c10">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> </span>)</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $1 &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $1 ) );<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>, <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>] <b>means </b>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> $2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[$2,<font color="Olive" title="b2">k2</font>] holds <br/><font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">k2</font> ) ) );<br/><div><i><font color="Green" title="E61">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">G</font> be    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Olive" title="b1">k1</font>]</span><br/><a NAME="E1:83_1_3"/>
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">G'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E3:83_1_3"/><i><font color="Green" title="E62">A9</font></i>: 
( <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">G'</font> &amp; <font color="Maroon" title="c13">G'</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c14">G'</font> = <font color="Maroon" title="c12">G</font> as   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="#E3:83_1_3"><i><font color="Green" title="E62">A9</font></i></a></i>;<br/><a NAME="E5:83_1_3"/><i><font color="Green" title="E63">A10</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">G'</font>
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green" title="E64">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_3_1"/><i><font color="Green" title="E65">A12</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/> <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c6">G2</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>
<br/>.= 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c14">G'</font>
<b>by </b><i><a class="txt" href="#E1:83_1_3_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E3:83_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:83_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="#E5:83_1_3"><i><font color="Green" title="E63">A10</font></i></a>, <a class="txt" href="#E1:83_1_3_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="glib_004.html#T41" target="_self" title="GLIB_004:th.41">Th36</a>, <a class="ref" href="glib_000.html#T89" title="GLIB_000:th.89">GLIB_000:89</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_3_2"/>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_3_2"/><b>then </b>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 <b>by </b><i><a class="txt" href="#E6:83_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:83_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G'</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></i>;<br/><a NAME="E4:83_1_3_2"/><b>then </b><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G'</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E7:83"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>;<br/><a NAME="E5:83_1_3_2"/>
<font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G'</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><b>hence </b><a NAME="E6:83_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:83_1_3_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E8:83_1_3"/><b>then </b><i><font color="Green" title="E65">A14</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">k3</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:83_1_3"/><i><font color="Green" title="E66">A15</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k3</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c15">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E8:83_1_3"><i><font color="Green" title="E65">A14</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_3_3"/>
<font color="Maroon" title="c15">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_3_3"/><b>then </b><i><font color="Green" title="E67">A16</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Init</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 <b>by </b><i><a class="txt" href="#E10:83_1_3"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="glib_004.html#D17" target="_self" title="GLIB_004:def.17">Def15</a></i>;<br/><a NAME="E3:83_1_3_3"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Init</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E4:83_1_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:83_1_3_3"><i><font color="Green" title="E67">A16</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:83_1_3"/><i><font color="Green" title="E67">A17</font></i>: 
<font color="Maroon" title="c16">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k3</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E14:83_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k3</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E13:83_1_3"><i><font color="Green" title="E67">A17</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E15:83_1_3"/><b>then </b><i><font color="Green" title="E68">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 <b>by </b><i><a class="txt" href="#E10:83_1_3"><i><font color="Green" title="E66">A15</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k2</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font> )</span><br/><b>assume </b><a NAME="E1:83_1_3_4"/>
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font></span><br/><a NAME="E2:83_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T34" target="_self" title="GLIB_004:th.34">Lm13</a></i>;<br/><b>hence </b><a NAME="E3:83_1_3_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">G'</font>
 <b>by </b><i><a class="txt" href="#E15:83_1_3"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E17:83_1_3"/>
 ex <font color="Olive" title="b1">k1</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Olive" title="b1">k1</font>]
 <b>by </b><i><a class="txt" href="#E10:83_1_3"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="#E13:83_1_3"><i><font color="Green" title="E67">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E62">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">G</font> be    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Olive" title="b1">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Olive" title="b2">k2</font>] holds <br/><font color="Olive" title="b1">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font></span><br/><b>let </b><font color="Maroon" title="c13">k1</font>, <font color="Maroon" title="c14">k2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Maroon" title="c13">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Maroon" title="c14">k2</font>] implies <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k2</font> )</span><br/><b>assume </b><a NAME="E1:83_1_4"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Maroon" title="c13">k1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">G</font>,<font color="Maroon" title="c14">k2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k2</font></span><br/><a NAME="E2:83_1_4"/><b>then </b>
( <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">k2</font> &amp; <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">k1</font> )
 ;<br/><a NAME="E3:83_1_4"/><b>then </b>
( <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">k2</font> &amp; <font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">k1</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:83_1_4"/>
<font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E8:83_1"/><i><font color="Green" title="E63">A21</font></i>: 
( <font color="Maroon" title="c11">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c11">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">X</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><a NAME="E1:83_1_5"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">x'</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:83_1_5"/><i><font color="Green" title="E64">A22</font></i>: 
( <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E4:83_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">y'</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E6:83_1_5"/><i><font color="Green" title="E65">A23</font></i>: 
( <font color="Maroon" title="c15">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><b>set </b><font color="Maroon" title="c16">CX</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c17">CY</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_5_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_5_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_5_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 <b>by </b><i><a class="txt" href="#E3:83_1_5"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E6:83_1_5"><i><font color="Green" title="E65">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_5_1_1_2"/><i><font color="Green" title="E66">A24</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c18">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:83_1_5_1_1_2"/><i><font color="Green" title="E67">A25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c18">k1</font>]
 <b>by </b><i><a class="txt" href="#E6:83_1"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:83_1_5_1_1_2"/><i><font color="Green" title="E68">A26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c19">k2</font>]
 <b>by </b><i><a class="txt" href="#E6:83_1"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_5_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_5_1_1_2_1_1"/>
( <font color="Maroon" title="c18">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c19">k2</font> or <font color="Maroon" title="c18">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_5_1_1_2_1_1_1"/><i><font color="Green" title="E69">A27</font></i>: 
<font color="Maroon" title="c18">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c19">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_5_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">z1</font>, <font color="Maroon" title="c21">z2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c20">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c21">z2</font>] implies <font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c21">z2</font> )</span><br/><b>assume </b><a NAME="E1:83_1_5_1_1_2_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c20">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c21">z2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c21">z2</font></span><br/><a NAME="E2:83_1_5_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k1</font> &amp; <font color="Maroon" title="c21">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k2</font> )
 <b>by </b><i><a class="txt" href="#E7:83_1"><i><font color="Green" title="E62">A19</font></i></a>, <a class="txt" href="#E3:83_1_5_1_1_2"><i><font color="Green" title="E67">A25</font></i></a>, <a class="txt" href="#E5:83_1_5_1_1_2"><i><font color="Green" title="E68">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:83_1_5_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c21">z2</font>
 <b>by </b><i><a class="txt" href="#E1:83_1_5_1_1_2_1_1_1"><i><font color="Green" title="E69">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:83_1_5_1_1_2_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 <b>by </b><i><a class="txt" href="#E3:83_1_5"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E6:83_1_5"><i><font color="Green" title="E65">A23</font></i></a>, <a class="txt" href="#E1:83_1_5_1_1_2"><i><font color="Green" title="E66">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_5_1_1_2_1_1_2"/><i><font color="Green" title="E69">A28</font></i>: 
<font color="Maroon" title="c18">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_5_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">z1</font>, <font color="Maroon" title="c21">z2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c20">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c21">z2</font>] implies <font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">z2</font> )</span><br/><b>assume </b><a NAME="E1:83_1_5_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c20">z1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c21">z2</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">z2</font></span><br/><a NAME="E2:83_1_5_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c20">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k1</font> &amp; <font color="Maroon" title="c21">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k2</font> )
 <b>by </b><i><a class="txt" href="#E7:83_1"><i><font color="Green" title="E62">A19</font></i></a>, <a class="txt" href="#E3:83_1_5_1_1_2"><i><font color="Green" title="E67">A25</font></i></a>, <a class="txt" href="#E5:83_1_5_1_1_2"><i><font color="Green" title="E68">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:83_1_5_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c20">z1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">z2</font>
 <b>by </b><i><a class="txt" href="#E1:83_1_5_1_1_2_1_1_2"><i><font color="Green" title="E69">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:83_1_5_1_1_2_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 <b>by </b><i><a class="txt" href="#E3:83_1_5"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E6:83_1_5"><i><font color="Green" title="E65">A23</font></i></a>, <a class="txt" href="#E1:83_1_5_1_1_2"><i><font color="Green" title="E66">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:83_1_5_1_1_2"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:83_1_5_1_1_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )</span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_5_1_1_3"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 <b>by </b><i><a class="txt" href="#E3:83_1_5"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="#E6:83_1_5"><i><font color="Green" title="E65">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:83_1_5"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c12">x</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:83_1"/><b>then </b><i><font color="Green" title="E64">A29</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>  holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> be    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>] )</span><br/><b>assume </b><a NAME="E1:83_1_6"/><i><font color="Green" title="E65">A30</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c13">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><a NAME="E2:83_1_6"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">x'</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E4:83_1_6"/><i><font color="Green" title="E66">A31</font></i>: 
( <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E5:83_1_6"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c16">y'</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E7:83_1_6"/><i><font color="Green" title="E67">A32</font></i>: 
( <font color="Maroon" title="c16">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E8:83_1_6"/>
<font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">z'</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E10:83_1_6"/><i><font color="Green" title="E68">A33</font></i>: 
( <font color="Maroon" title="c17">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">z</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><b>set </b><font color="Maroon" title="c18">CX</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c19">CY</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c20">CZ</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_6_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">kx</font>, <font color="Olive" title="b2">ky</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x'</font>,<font color="Olive" title="b1">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b2">ky</font>] holds <br/><font color="Olive" title="b1">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">ky</font> ) ) )
 </span><b>by </b><i><a class="txt" href="#E1:83_1_6"><i><font color="Green" title="E65">A30</font></i></a>, <a class="txt" href="#E4:83_1_6"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="#E7:83_1_6"><i><font color="Green" title="E67">A32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_1"/><i><font color="Green" title="E69">A34</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_6_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_6_1_1_1_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">ky</font>, <font color="Olive" title="b2">kz</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b1">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Olive" title="b2">kz</font>] holds <br/><font color="Olive" title="b1">ky</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">kz</font> ) ) )
 </span><b>by </b><i><a class="txt" href="#E1:83_1_6"><i><font color="Green" title="E65">A30</font></i></a>, <a class="txt" href="#E7:83_1_6"><i><font color="Green" title="E67">A32</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_1_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_6_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 <b>by </b><i><a class="txt" href="#E4:83_1_6"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_1"><i><font color="Green" title="E69">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_1_1_1_2"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">ky</font>, <font color="Olive" title="b2">kz</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b1">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Olive" title="b2">kz</font>] holds <br/><font color="Olive" title="b1">ky</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">kz</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_6_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 <b>by </b><i><a class="txt" href="#E4:83_1_6"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_1"><i><font color="Green" title="E69">A34</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:83_1_6_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_2"/><i><font color="Green" title="E69">A35</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">kx</font>, <font color="Olive" title="b2">ky</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x'</font>,<font color="Olive" title="b1">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b2">ky</font>] holds <br/><font color="Olive" title="b1">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">ky</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_6_1_1_2_1_1"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> or (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">ky</font>, <font color="Olive" title="b2">kz</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b1">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Olive" title="b2">kz</font>] holds <br/><font color="Olive" title="b1">ky</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">kz</font> ) ) )
 </span><b>by </b><i><a class="txt" href="#E1:83_1_6"><i><font color="Green" title="E65">A30</font></i></a>, <a class="txt" href="#E7:83_1_6"><i><font color="Green" title="E67">A32</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_2_1_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:83_1_6_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 <b>by </b><i><a class="txt" href="#E4:83_1_6"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_2"><i><font color="Green" title="E69">A35</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_6_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_6_1_1_2_1_1_2"/><i><font color="Green" title="E70">A36</font></i>: 
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c16">y'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c17">z'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">ky</font>, <font color="Olive" title="b2">kz</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">y'</font>,<font color="Olive" title="b1">ky</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Olive" title="b2">kz</font>] holds <br/><font color="Olive" title="b1">ky</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">kz</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c21">zx</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>consider </b><font color="Maroon" title="c22">zy</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:83_1_6_1_1_2_1_1_2"/><i><font color="Green" title="E71">A37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c22">zy</font>]
 <b>by </b><i><a class="txt" href="#E6:83_1"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c23">zz</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_6_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">kx</font>, <font color="Maroon" title="c25">kz</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c24">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c25">kz</font>] implies <font color="Maroon" title="c24">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">kz</font> )</span><br/><b>assume </b><a NAME="E1:83_1_6_1_1_2_1_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x'</font>,<font color="Maroon" title="c24">kx</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">z'</font>,<font color="Maroon" title="c25">kz</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c24">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">kz</font></span><br/><a NAME="E2:83_1_6_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c24">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c22">zy</font> &amp; <font color="Maroon" title="c22">zy</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">kz</font> )
 <b>by </b><i><a class="txt" href="#E7:83_1_6"><i><font color="Green" title="E67">A32</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_2"><i><font color="Green" title="E69">A35</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_2_1_1_2"><i><font color="Green" title="E70">A36</font></i></a>, <a class="txt" href="#E4:83_1_6_1_1_2_1_1_2"><i><font color="Green" title="E71">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:83_1_6_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c24">kx</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c25">kz</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:83_1_6_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 <b>by </b><i><a class="txt" href="#E4:83_1_6"><i><font color="Green" title="E66">A31</font></i></a>, <a class="txt" href="#E10:83_1_6"><i><font color="Green" title="E68">A33</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_2"><i><font color="Green" title="E69">A35</font></i></a>, <a class="txt" href="#E1:83_1_6_1_1_2_1_1_2"><i><font color="Green" title="E70">A36</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:83_1_6_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:83_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c14">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:83_1"/><b>then </b><i><font color="Green" title="E65">A38</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c12">M</font> being    <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font><b> such that </b><br/><a NAME="E14:83_1"/><i><font color="Green" title="E66">A39</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="glib_004.html#M1" title="GLIB_004:mode.1">Element</a> of <font color="Maroon" title="c11">X</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">M</font>,<font color="Olive" title="b1">y</font>] ) )
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S4" title="CQC_SIM1:sch.4">CQC_SIM1:sch 4</a>(<a class="txt" href="#E8:83_1"><i><font color="Green" title="E63">A21</font></i></a>, <a class="txt" href="#E10:83_1"><i><font color="Green" title="E64">A29</font></i></a>, <a class="txt" href="#E12:83_1"><i><font color="Green" title="E65">A38</font></i></a>);<br/></i><b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> <b> such that </b><br/><a NAME="E16:83_1"/><i><font color="Green" title="E67">A40</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c13">x</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="txt" href="#E14:83_1"><i><font color="Green" title="E66">A39</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c12">M</font> as   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="#E16:83_1"><i><font color="Green" title="E67">A40</font></i></a></i>;<br/><a NAME="E18:83_1"/><i><font color="Green" title="E68">A41</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><div><i><font color="Green" title="E69">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_7"/><i><font color="Green" title="E70">A43</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/> <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c6">G2</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>
<br/>.= 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c14">M</font>
<b>by </b><i><a class="txt" href="#E1:83_1_7"><i><font color="Green" title="E70">A43</font></i></a>, <a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>
;<br/><b>hence </b><a NAME="E3:83_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:83_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="#E18:83_1"><i><font color="Green" title="E68">A41</font></i></a>, <a class="txt" href="#E1:83_1_7"><i><font color="Green" title="E70">A43</font></i></a>, <a class="ref" href="glib_004.html#T41" target="_self" title="GLIB_004:th.41">Th36</a>, <a class="ref" href="glib_000.html#T89" title="GLIB_000:th.89">GLIB_000:89</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> $1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_8"/>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_8"/><b>then </b>
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E19:83_1"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:83_1_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></i>;<br/><a NAME="E4:83_1_8"/><b>then </b><i><font color="Green" title="E70">A44</font></i>: 
<font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E7:83"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>;<br/><a NAME="E5:83_1_8"/>
<font color="Maroon" title="c6">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">M</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><b>hence </b><a NAME="E6:83_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:83_1_8"><i><font color="Green" title="E70">A44</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E21:83_1"/><b>then </b><i><font color="Green" title="E70">A45</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E23:83_1"/><i><font color="Green" title="E71">A46</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c15">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E21:83_1"><i><font color="Green" title="E70">A45</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_9"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_9"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_9"/><b>then </b><i><font color="Green" title="E72">A47</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Init</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E23:83_1"><i><font color="Green" title="E71">A46</font></i></a>, <a class="ref" href="glib_004.html#D17" target="_self" title="GLIB_004:def.17">Def15</a></i>;<br/><a NAME="E3:83_1_9"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Init</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E4:83_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:83_1_9"><i><font color="Green" title="E72">A47</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c16">k1o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:83_1"/><i><font color="Green" title="E72">A48</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k1o</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>set </b><font color="Maroon" title="c17">Gk1b</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font>;<br/><b>set </b><font color="Maroon" title="c18">Gk</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font>;<br/><a NAME="E27:83_1"/><i><font color="Green" title="E73">A49</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E26:83_1"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="glib_004.html#D17" target="_self" title="GLIB_004:def.17">Def15</a></i>;<br/><a NAME="E28:83_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">k1o</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E26:83_1"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E29:83_1"/><b>then </b><i><font color="Green" title="E74">A50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E23:83_1"><i><font color="Green" title="E71">A46</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c19">Next</font> =  <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c20">ep</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>;<br/><a NAME="E30:83_1"/><i><font color="Green" title="E75">A51</font></i>: 
 <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E23:83_1"><i><font color="Green" title="E71">A46</font></i></a>, <a class="txt" href="#E27:83_1"><i><font color="Green" title="E73">A49</font></i></a>, <a class="txt" href="#E29:83_1"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def14</a></i>;<br/><a NAME="E31:83_1"/><b>then </b><i><font color="Green" title="E76">A52</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c21">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E33:83_1"/><i><font color="Green" title="E77">A53</font></i>: 
( not <font color="Maroon" title="c21">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E27:83_1"><i><font color="Green" title="E73">A49</font></i></a>, <a class="txt" href="#E30:83_1"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="glib_004.html#T28" target="_self" title="GLIB_004:th.28">Lm6</a></i>;<br/><a NAME="E34:83_1"/><i><font color="Green" title="E78">A54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E33:83_1"><i><font color="Green" title="E77">A53</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E35:83_1"/><b>then </b><i><font color="Green" title="E79">A55</font></i>: 
not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E23:83_1"><i><font color="Green" title="E71">A46</font></i></a>, <a class="txt" href="#E29:83_1"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E36:83_1"/><b>then </b><i><font color="Green" title="E80">A56</font></i>: 
not  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>set </b><font color="Maroon" title="c22">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/><a NAME="E37:83_1"/><i><font color="Green" title="E81">A58</font></i>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <b>by </b><i><a class="ref" href="glib_004.html#T30" target="_self" title="GLIB_004:th.30">Lm9</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">Mep</font> being   <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G1</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>set </b><font color="Maroon" title="c25">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c26">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E40:83_1"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="txt" href="#E31:83_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E41:83_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E42:83_1"/><b>then </b>
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T37" title="GLIB_000:th.37">GLIB_000:37</a></i>;<br/><a NAME="E43:83_1"/><b>then </b><i><font color="Green" title="E82">A59</font></i>: 
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c24">Mep</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E44:83_1"/><b>then </b>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c24">Mep</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">M'</font> = <font color="Maroon" title="c14">M</font> as  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c24">Mep</font> <b>by </b><i><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></i>;<br/><a NAME="E46:83_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E47:83_1"/><b>then </b><i><font color="Green" title="E83">A60</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E48:83_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c24">Mep</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon" title="c29">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c14">M</font> <b>by </b><i><a class="txt" href="#E47:83_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>consider </b><font color="Maroon" title="c30">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c27">M'</font><b> such that </b><br/><a NAME="E51:83_1"/><i><font color="Green" title="E84">A61</font></i>: 
<font color="Maroon" title="c30">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c28">v1</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c31">PW</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c30">W</font>;<br/><a NAME="E53:83_1"/><i><font color="Green" title="E85">A62</font></i>: 
<font color="Maroon" title="c31">PW</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c28">v1</font>
 <b>by </b><i><a class="txt" href="#E51:83_1"><i><font color="Green" title="E84">A61</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><a NAME="E54:83_1"/><i><font color="Green" title="E86">A63</font></i>: 
<font color="Maroon" title="c31">PW</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c32">P</font> = <font color="Maroon" title="c31">PW</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c24">Mep</font> <b>by </b><i><a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></i>;<br/><a NAME="E56:83_1"/><i><font color="Green" title="E87">A64</font></i>: 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E54:83_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="glib_001.html#T111" title="GLIB_001:th.111">GLIB_001:111</a></i>;<br/><a NAME="E57:83_1"/><i><font color="Green" title="E88">A65</font></i>: 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c28">v1</font>
 <b>by </b><i><a class="txt" href="#E53:83_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><a NAME="E58:83_1"/>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E59:83_1"/><b>then </b><i><font color="Green" title="E89">A66</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="#E34:83_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E60:83_1"/><i><font color="Green" title="E90">A67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K17" title="GLIB_004:func.17">PRIM:MST</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T42" target="_self" title="GLIB_004:th.42">Th52</a></i>;<br/><a NAME="E61:83_1"/><b>then </b><i><font color="Green" title="E91">A68</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E59:83_1"><i><font color="Green" title="E89">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T52" title="ZFMISC_1:th.52">ZFMISC_1:52</a></i>;<br/><a NAME="E62:83_1"/><i><font color="Green" title="E92">A69</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c28">v1</font>,<font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c24">Mep</font>
 <b>by </b><i><a class="txt" href="#E47:83_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><a NAME="E63:83_1"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c28">v1</font>,<font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><a NAME="E64:83_1"/><b>then </b><i><font color="Green" title="E93">A70</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c28">v1</font>,<font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c8">G2'</font>
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E59:83_1"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="#E60:83_1"><i><font color="Green" title="E90">A67</font></i></a>, <a class="ref" href="glib_000.html#T76" title="GLIB_000:th.76">GLIB_000:76</a></i>;<br/><a NAME="E65:83_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">v2</font> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">v2</font> ) )
 <b>by </b><i><a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><a NAME="E66:83_1"/><b>then </b>
<font color="Maroon" title="c28">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">v2</font>
 <b>by </b><i><a class="txt" href="#E6:83"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="#E59:83_1"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="#E60:83_1"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="#E64:83_1"><i><font color="Green" title="E93">A70</font></i></a>, <a class="ref" href="glib_000.html#D20" title="GLIB_000:def.20">GLIB_000:def 20</a></i>;<br/><a NAME="E67:83_1"/><b>then </b>
<font color="Maroon" title="c28">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E68:83_1"/><b>then </b>
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E69:83_1"/><b>then </b><i><font color="Green" title="E94">A71</font></i>: 
not <font color="Maroon" title="c32">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><a NAME="E70:83_1"/><i><font color="Green" title="E95">A72</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c24">Mep</font>
 <b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="txt" href="#E62:83_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E71:83_1"/><i><font color="Green" title="E96">A73</font></i>: 
not  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E35:83_1"><i><font color="Green" title="E79">A55</font></i></a>, <a class="txt" href="#E56:83_1"><i><font color="Green" title="E87">A64</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c33">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">n</font> &amp; <font color="Maroon" title="c33">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> implies  not <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">v2</font> )</span><br/><b>assume </b><a NAME="E1:83_1_10"/><i><font color="Green" title="E97">A74</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">n</font> &amp; <font color="Maroon" title="c33">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> &amp; <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">v2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c29">v2</font> = 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>
;<br/><a NAME="E3:83_1_10"/><b>then </b>
<font color="Maroon" title="c33">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font>
 <b>by </b><i><a class="txt" href="#E1:83_1_10"><i><font color="Green" title="E97">A74</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>then </b><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<font color="Maroon" title="c29">v2</font>
<b>by </b><i><a class="txt" href="#E1:83_1_10"><i><font color="Green" title="E97">A74</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/><a NAME="E5:83_1_10"/><b>then </b><i><font color="Green" title="E98">A75</font></i>: 
<font color="Maroon" title="c32">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">PM</font> = <font color="Maroon" title="c32">P</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c14">M</font> ;<br/><a NAME="E7:83_1_10"/>
( <font color="Maroon" title="c34">PM</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; <font color="Maroon" title="c34">PM</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c34">PM</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )
 <b>by </b><i><a class="txt" href="#E1:83_1_10"><i><font color="Green" title="E97">A74</font></i></a>, <a class="txt" href="#E5:83_1_10"><i><font color="Green" title="E98">A75</font></i></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a>, <a class="ref" href="glib_001.html#T177" title="GLIB_001:th.177">GLIB_001:177</a></i>;<br/><a NAME="E8:83_1_10"/><b>then </b>
<font color="Maroon" title="c34">PM</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><b>hence </b><a NAME="E9:83_1_10"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_002.html#D2" title="GLIB_002:def.2">GLIB_002:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E73:83_1"/><b>then </b><i><font color="Green" title="E97">A76</font></i>: 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="#E69:83_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="txt" href="#E71:83_1"><i><font color="Green" title="E96">A73</font></i></a>, <a class="ref" href="glib_001.html#T151" title="GLIB_001:th.151">GLIB_001:151</a></i>;<br/><b>set </b><font color="Maroon" title="c33">C</font> = <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E74:83_1"/><i><font color="Green" title="E98">A77</font></i>: 
not <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T133" title="GLIB_001:th.133">GLIB_001:133</a></i>;<br/><a NAME="E75:83_1"/>
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c29">v2</font>,<font color="Maroon" title="c29">v2</font>
 <b>by </b><i><a class="txt" href="#E57:83_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="txt" href="#E62:83_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="ref" href="glib_001.html#T67" title="GLIB_001:th.67">GLIB_001:67</a></i>;<br/><a NAME="E76:83_1"/><b>then </b>
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T120" title="GLIB_001:th.120">GLIB_001:120</a></i>;<br/><a NAME="E77:83_1"/><b>then </b><i><font color="Green" title="E99">A78</font></i>: 
<font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="txt" href="#E73:83_1"><i><font color="Green" title="E97">A76</font></i></a>, <a class="txt" href="#E74:83_1"><i><font color="Green" title="E98">A77</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><a NAME="E78:83_1"/><i><font color="Green" title="E100">A79</font></i>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="#E30:83_1"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="glib_004.html#D13" target="_self" title="GLIB_004:def.13">Def12</a></i>;<br/><a NAME="E79:83_1"/><i><font color="Green" title="E101">A80</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">v2</font> )
 <b>by </b><i><a class="txt" href="#E47:83_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>;<br/><a NAME="E80:83_1"/><i><font color="Green" title="E102">A81</font></i>: 
( <font color="Maroon" title="c28">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> &amp; <font color="Maroon" title="c29">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="#E53:83_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_11"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_11_1"/>
( ( <font color="Maroon" title="c28">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c29">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> ) or ( <font color="Maroon" title="c29">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c28">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="#E78:83_1"><i><font color="Green" title="E100">A79</font></i></a>, <a class="txt" href="#E79:83_1"><i><font color="Green" title="E101">A80</font></i></a>, <a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_11_1_1"><b>suppose </b></a><a NAME="E1:83_1_11_1_1"/><i><font color="Green" title="E103">A82</font></i>: 
( <font color="Maroon" title="c28">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c29">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">em</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Olive" title="b1">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( not $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> &amp; <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  );<br/><a NAME="E2:83_1_11_1_1"/><i><font color="Green" title="E104">A83</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="#E80:83_1"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="#E1:83_1_11_1_1"><i><font color="Green" title="E103">A82</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c34">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:83_1_11_1_1"/><i><font color="Green" title="E105">A84</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c34">m</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c34">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E2:83_1_11_1_1"><i><font color="Green" title="E104">A83</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c35">m</font> = <font color="Maroon" title="c34">m</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:83_1_11_1_1"/><i><font color="Green" title="E106">A85</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">m</font> &amp; <font color="Maroon" title="c35">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> )
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E7:83_1_11_1_1"/>
<font color="Maroon" title="c35">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="#E80:83_1"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="#E1:83_1_11_1_1"><i><font color="Green" title="E103">A82</font></i></a>, <a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E8:83_1_11_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">m</font>
 <b>by </b><i><a class="txt" href="#E6:83_1_11_1_1"><i><font color="Green" title="E106">A85</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:83_1_11_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">m2k</font> = <font color="Maroon" title="c35">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E11:83_1_11_1_1"/><i><font color="Green" title="E107">A86</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E12:83_1_11_1_1"/><b>then </b><i><font color="Green" title="E108">A87</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font>
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:83_1_11_1_1"/><b>then </b><i><font color="Green" title="E109">A88</font></i>: 
not <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="txt" href="#E11:83_1_11_1_1"><i><font color="Green" title="E107">A86</font></i></a></i>;<br/><a NAME="E14:83_1_11_1_1"/><i><font color="Green" title="E110">A89</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">m</font>
 ;<br/><b>set </b><font color="Maroon" title="c37">em</font> = <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c38">em</font> = <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><a NAME="E15:83_1_11_1_1"/><i><font color="Green" title="E111">A90</font></i>: 
<font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E12:83_1_11_1_1"><i><font color="Green" title="E108">A87</font></i></a>, <a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a></i>;<br/><a NAME="E16:83_1_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T112" title="GLIB_001:th.112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:83_1_11_1_1"/>
<font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E15:83_1_11_1_1"><i><font color="Green" title="E111">A90</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><b>consider </b><font color="Maroon" title="c39">i</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:83_1_11_1_1"/><i><font color="Green" title="E112">A91</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> &amp; <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">em</font> )
 <b>by </b><i><a class="txt" href="#E15:83_1_11_1_1"><i><font color="Green" title="E111">A90</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><a NAME="E20:83_1_11_1_1"/>
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">P</font>
 <b>by </b><i><a class="txt" href="#E19:83_1_11_1_1"><i><font color="Green" title="E112">A91</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:83_1_11_1_1"/><b>then </b><i><font color="Green" title="E113">A92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">em</font>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="txt" href="#E19:83_1_11_1_1"><i><font color="Green" title="E112">A91</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/><a NAME="E22:83_1_11_1_1"/><i><font color="Green" title="E114">A93</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/><a NAME="E23:83_1_11_1_1"/><i><font color="Green" title="E115">A94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/><a NAME="E24:83_1_11_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E25:83_1_11_1_1"/><b>then </b><i><font color="Green" title="E116">A95</font></i>: 
<font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E19:83_1_11_1_1"><i><font color="Green" title="E112">A91</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E26:83_1_11_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E27:83_1_11_1_1"/>
<font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E73:83_1"><i><font color="Green" title="E97">A76</font></i></a>, <a class="txt" href="#E19:83_1_11_1_1"><i><font color="Green" title="E112">A91</font></i></a>, <a class="txt" href="#E21:83_1_11_1_1"><i><font color="Green" title="E113">A92</font></i></a>, <a class="txt" href="#E22:83_1_11_1_1"><i><font color="Green" title="E114">A93</font></i></a>, <a class="txt" href="#E23:83_1_11_1_1"><i><font color="Green" title="E115">A94</font></i></a>, <a class="txt" href="#E25:83_1_11_1_1"><i><font color="Green" title="E116">A95</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font></span><br/><a NAME="E28:83_1_11_1_1"/><i><font color="Green" title="E117">A97</font></i>: 
<font color="Maroon" title="c38">em</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font>,<font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">m</font>,<font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E12:83_1_11_1_1"><i><font color="Green" title="E108">A87</font></i></a>, <a class="txt" href="#E14:83_1_11_1_1"><i><font color="Green" title="E110">A89</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E29:83_1_11_1_1"/><b>then </b>
<font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T16" title="GLIB_000:th.16">GLIB_000:16</a></i>;<br/><a NAME="E30:83_1_11_1_1"/><b>then </b>
<font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E13:83_1_11_1_1"><i><font color="Green" title="E109">A88</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E31:83_1_11_1_1"/>
<font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_1"><i><font color="Green" title="E105">A84</font></i></a>, <a class="txt" href="#E28:83_1_11_1_1"><i><font color="Green" title="E117">A97</font></i></a>, <a class="ref" href="glib_000.html#T20" title="GLIB_000:th.20">GLIB_000:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_11_1_2"><b>suppose </b></a><a NAME="E1:83_1_11_1_2"/><i><font color="Green" title="E103">A98</font></i>: 
( <font color="Maroon" title="c29">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c28">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">em</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Olive" title="b1">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( not $1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> &amp; <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> );<br/><a NAME="E2:83_1_11_1_2"/><i><font color="Green" title="E104">A99</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="#E80:83_1"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="#E1:83_1_11_1_2"><i><font color="Green" title="E103">A98</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c34">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:83_1_11_1_2"/><i><font color="Green" title="E105">A100</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c34">m</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c34">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E2:83_1_11_1_2"><i><font color="Green" title="E104">A99</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c35">m</font> = <font color="Maroon" title="c34">m</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:83_1_11_1_2"/><i><font color="Green" title="E106">A101</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">m</font> &amp; <font color="Maroon" title="c35">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> )
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E7:83_1_11_1_2"/>
<font color="Maroon" title="c35">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="#E80:83_1"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="#E1:83_1_11_1_2"><i><font color="Green" title="E103">A98</font></i></a>, <a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E8:83_1_11_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">m</font>
 <b>by </b><i><a class="txt" href="#E6:83_1_11_1_2"><i><font color="Green" title="E106">A101</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:83_1_11_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c36">m2k</font> = <font color="Maroon" title="c35">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E11:83_1_11_1_2"/><i><font color="Green" title="E107">A102</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E12:83_1_11_1_2"/><b>then </b><i><font color="Green" title="E108">A103</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font>
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E109">A104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_11_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_11_1_2_1"/><i><font color="Green" title="E110">A105</font></i>: 
not <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_11_1_2_1"/>
<font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="#E37:83_1"><i><font color="Green" title="E81">A58</font></i></a>, <a class="txt" href="#E12:83_1_11_1_2"><i><font color="Green" title="E108">A103</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></i>;<br/><a NAME="E3:83_1_11_1_2_1"/><b>then </b>
<font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:83_1_11_1_2_1"><i><font color="Green" title="E110">A105</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E4:83_1_11_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="txt" href="#E11:83_1_11_1_2"><i><font color="Green" title="E107">A102</font></i></a>, <a class="txt" href="#E12:83_1_11_1_2"><i><font color="Green" title="E108">A103</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E14:83_1_11_1_2"/><i><font color="Green" title="E110">A106</font></i>: 
<font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">m</font>
 ;<br/><b>set </b><font color="Maroon" title="c37">em</font> = <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c38">em</font> = <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">m2k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><a NAME="E15:83_1_11_1_2"/><i><font color="Green" title="E111">A107</font></i>: 
<font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E12:83_1_11_1_2"><i><font color="Green" title="E108">A103</font></i></a>, <a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a></i>;<br/><a NAME="E16:83_1_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T112" title="GLIB_001:th.112">GLIB_001:112</a></i>;<br/><b>hence </b><a NAME="E17:83_1_11_1_2"/>
<font color="Maroon" title="c38">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E15:83_1_11_1_2"><i><font color="Green" title="E111">A107</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )</span><br/><b>consider </b><font color="Maroon" title="c39">i</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:83_1_11_1_2"/><i><font color="Green" title="E112">A108</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">i</font> &amp; <font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font> &amp; <font color="Maroon" title="c32">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">em</font> )
 <b>by </b><i><a class="txt" href="#E15:83_1_11_1_2"><i><font color="Green" title="E111">A107</font></i></a>, <a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><a NAME="E20:83_1_11_1_2"/>
<font color="Maroon" title="c39">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">P</font>
 <b>by </b><i><a class="txt" href="#E19:83_1_11_1_2"><i><font color="Green" title="E112">A108</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:83_1_11_1_2"/><b>then </b><i><font color="Green" title="E113">A109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">em</font>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="txt" href="#E19:83_1_11_1_2"><i><font color="Green" title="E112">A108</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/><a NAME="E22:83_1_11_1_2"/><i><font color="Green" title="E114">A110</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></i>;<br/><a NAME="E23:83_1_11_1_2"/><i><font color="Green" title="E115">A111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E70:83_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></i>;<br/><a NAME="E24:83_1_11_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E25:83_1_11_1_2"/><b>then </b><i><font color="Green" title="E116">A112</font></i>: 
<font color="Maroon" title="c39">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E19:83_1_11_1_2"><i><font color="Green" title="E112">A108</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E26:83_1_11_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E27:83_1_11_1_2"/>
<font color="Maroon" title="c38">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E73:83_1"><i><font color="Green" title="E97">A76</font></i></a>, <a class="txt" href="#E19:83_1_11_1_2"><i><font color="Green" title="E112">A108</font></i></a>, <a class="txt" href="#E21:83_1_11_1_2"><i><font color="Green" title="E113">A109</font></i></a>, <a class="txt" href="#E22:83_1_11_1_2"><i><font color="Green" title="E114">A110</font></i></a>, <a class="txt" href="#E23:83_1_11_1_2"><i><font color="Green" title="E115">A111</font></i></a>, <a class="txt" href="#E25:83_1_11_1_2"><i><font color="Green" title="E116">A112</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font></span><br/><a NAME="E28:83_1_11_1_2"/>
<font color="Maroon" title="c38">em</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">m2k</font>,<font color="Maroon" title="c31">PW</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">m</font>,<font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E12:83_1_11_1_2"><i><font color="Green" title="E108">A103</font></i></a>, <a class="txt" href="#E14:83_1_11_1_2"><i><font color="Green" title="E110">A106</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><b>hence </b><a NAME="E29:83_1_11_1_2"/>
<font color="Maroon" title="c38">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E37:83_1"><i><font color="Green" title="E81">A58</font></i></a>, <a class="txt" href="#E4:83_1_11_1_2"><i><font color="Green" title="E105">A100</font></i></a>, <a class="txt" href="#E13:83_1_11_1_2"><i><font color="Green" title="E109">A104</font></i></a>, <a class="ref" href="glib_000.html#T20" title="GLIB_000:th.20">GLIB_000:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c34">em</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E83:83_1"/><i><font color="Green" title="E103">A114</font></i>: 
( <font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c34">em</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c34">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c14">M</font> )
 ;<br/><a NAME="E84:83_1"/><i><font color="Green" title="E104">A115</font></i>: 
<font color="Maroon" title="c34">em</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c23">V</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c23">V</font>,<font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="#E37:83_1"><i><font color="Green" title="E81">A58</font></i></a>, <a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="ref" href="glib_000.html#T75" title="GLIB_000:th.75">GLIB_000:75</a></i>;<br/><a NAME="E85:83_1"/><b>then </b><i><font color="Green" title="E105">A116</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font>
 <b>by </b><i><a class="txt" href="#E30:83_1"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="glib_004.html#D13" target="_self" title="GLIB_004:def.13">Def12</a></i>;<br/><b>consider </b><font color="Maroon" title="c35">M2</font> being  <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a> <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a> <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c24">Mep</font>,<font color="Maroon" title="c34">em</font>;<br/><b>reconsider </b><font color="Maroon" title="c36">M2</font> = <font color="Maroon" title="c35">M2</font> as   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="ref" href="glib_003.html#T16" title="GLIB_003:th.16">GLIB_003:16</a></i>;<br/><a NAME="E88:83_1"/>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c24">Mep</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font> )
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c37">M'</font> = <font color="Maroon" title="c14">M</font> as  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c24">Mep</font> <b>by </b><i><a class="ref" href="glib_000.html#T47" title="GLIB_000:th.47">GLIB_000:47</a></i>;<br/><a NAME="E90:83_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c37">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c24">Mep</font>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><a NAME="E91:83_1"/><b>then </b>
<font color="Maroon" title="c37">M'</font> is <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><a NAME="E92:83_1"/><b>then </b>
<font color="Maroon" title="c24">Mep</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T23" title="GLIB_002:th.23">GLIB_002:23</a></i>;<br/><a NAME="E93:83_1"/><b>then </b><i><font color="Green" title="E106">A117</font></i>: 
<font color="Maroon" title="c36">M2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="#E77:83_1"><i><font color="Green" title="E99">A78</font></i></a>, <a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="ref" href="glib_002.html#T5" title="GLIB_002:th.5">GLIB_002:5</a></i>;<br/><a NAME="E94:83_1"/><i><font color="Green" title="E107">A118</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c36">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="glib_000.html#T54" title="GLIB_000:th.54">GLIB_000:54</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_12"><b>now </b></a><div class="add"><a NAME="E1:83_1_12"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>then </b><i><font color="Green" title="E108">A119</font></i>: <font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a>  = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E94:83_1"><i><font color="Green" title="E107">A118</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="#E36:83_1"><i><font color="Green" title="E80">A56</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>
<br/>.= 
<font color="Maroon" title="c14">M</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> 

;<br/><font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="glib_000.html#T56" title="GLIB_000:th.56">GLIB_000:56</a></i>
<br/>.= 
<font color="Maroon" title="c14">M</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> 
<b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>
;<br/><b>hence </b><a NAME="E4:83_1_12"/>
<font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E2:83_1_12"><i><font color="Green" title="E108">A119</font></i></a>, <a class="ref" href="glib_002.html#T46" title="GLIB_002:th.46">GLIB_002:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E96:83_1"/><b>then </b><i><font color="Green" title="E108">A120</font></i>: 
<font color="Maroon" title="c36">M2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="#E93:83_1"><i><font color="Green" title="E106">A117</font></i></a>, <a class="ref" href="glib_002.html#T47" title="GLIB_002:th.47">GLIB_002:47</a></i>;<br/><b>set </b><font color="Maroon" title="c38">M2'</font> = <font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ;<br/><a NAME="E97:83_1"/><i><font color="Green" title="E109">A121</font></i>: 
( <font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c36">M2</font> &amp;  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c36">M2</font> )
 <b>by </b><i><a class="txt" href="glib_004.html#E17"><i><font color="Green" title="E13">Lm2</font></i></a></i>;<br/><a NAME="E98:83_1"/><b>then </b><i><font color="Green" title="E110">A122</font></i>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E94:83_1"><i><font color="Green" title="E107">A118</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c39">M2'</font> = <font color="Maroon" title="c36">M2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.strict</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  as   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="#E97:83_1"><i><font color="Green" title="E109">A121</font></i></a>, <a class="ref" href="glib_003.html#T15" title="GLIB_003:th.15">GLIB_003:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_13"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:83_1_13"/>
<font color="Maroon" title="c39">M2'</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 <b>by </b><i><a class="txt" href="#E96:83_1"><i><font color="Green" title="E108">A120</font></i></a>, <a class="txt" href="glib_004.html#E17"><i><font color="Green" title="E13">Lm2</font></i></a>, <a class="ref" href="glib_002.html#T48" title="GLIB_002:th.48">GLIB_002:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c39">M2'</font> is <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a></span><br/> <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c39">M2'</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c36">M2</font>
<b>by </b><i><a class="txt" href="#E97:83_1"><i><font color="Green" title="E109">A121</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="glib_000.html#T56" title="GLIB_000:th.56">GLIB_000:56</a></i>
;<br/><b>hence </b><a NAME="E3:83_1_13"/>
<font color="Maroon" title="c39">M2'</font> is <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c40">M2'</font> = <font color="Maroon" title="c39">M2'</font> as  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_14"><b>now </b></a><div class="add"><a NAME="E1:83_1_14"/><i><font color="Green" title="E111">A123</font></i>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T37" title="GLIB_000:th.37">GLIB_000:37</a></i>;<br/><a NAME="E2:83_1_14"/><b>then </b><i><font color="Green" title="E112">A124</font></i>: 
(  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> )
 ;<br/><a NAME="E3:83_1_14"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D35" title="GLIB_000:def.35">GLIB_000:def 35</a></i>;<br/><a NAME="E4:83_1_14"/><b>then </b>
<font color="Maroon" title="c14">M</font> is   <a href="glib_003.html#NM15" title="GLIB_003:NM.15">inducedWSubgraph</a> of <font color="Maroon" title="c1">G1</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>, <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E1:83_1_14"><i><font color="Green" title="E111">A123</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><a NAME="E5:83_1_14"/><b>then </b><i><font color="Green" title="E113">A126</font></i>: 
<font color="Maroon" title="c24">Mep</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E31:83_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="#E36:83_1"><i><font color="Green" title="E80">A56</font></i></a>, <a class="txt" href="#E1:83_1_14"><i><font color="Green" title="E111">A123</font></i></a>, <a class="txt" href="#E2:83_1_14"><i><font color="Green" title="E112">A124</font></i></a>, <a class="ref" href="glib_004.html#T11" target="_self" title="GLIB_004:th.11">Th34</a></i>;<br/><a NAME="E6:83_1_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c36">M2</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Mep</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="ref" href="glib_004.html#T10" target="_self" title="GLIB_004:th.10">Th33</a></i>;<br/><a NAME="E7:83_1_14"/><b>then </b><i><font color="Green" title="E114">A127</font></i>: 
<font color="Maroon" title="c36">M2</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">Mep</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span>
 ;<br/><a NAME="E8:83_1_14"/>
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c24">Mep</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>;<br/><a NAME="E9:83_1_14"/><b>then </b>
<font color="Maroon" title="c36">M2</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="txt" href="#E5:83_1_14"><i><font color="Green" title="E113">A126</font></i></a>, <a class="txt" href="#E7:83_1_14"><i><font color="Green" title="E114">A127</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E10:83_1_14"/>
<font color="Maroon" title="c40">M2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E97:83_1"><i><font color="Green" title="E109">A121</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E103:83_1"/><b>then </b><i><font color="Green" title="E111">A128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">M2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c34">em</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">M</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E85:83_1"><i><font color="Green" title="E105">A116</font></i></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c41">G3</font> be  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c40">M2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> </span><br/><a NAME="E1:83_1_15"/>
<font color="Maroon" title="c14">M</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D19" target="_self" title="GLIB_004:def.19">Def17</a></i>;<br/><b>hence </b><a NAME="E2:83_1_15"/>
<font color="Maroon" title="c40">M2'</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">G3</font> <a href="glib_004.html#K4" title="GLIB_004:func.4">.cost()</a> 
 <b>by </b><i><a class="txt" href="#E103:83_1"><i><font color="Green" title="E111">A128</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c41">M2'</font> = <font color="Maroon" title="c40">M2'</font> as   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="ref" href="glib_004.html#D19" target="_self" title="GLIB_004:def.19">Def17</a></i>;<br/><b>set </b><font color="Maroon" title="c42">MG2'</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c43">MG2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>;<br/><a NAME="E106:83_1"/><i><font color="Green" title="E112">A129</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E98:83_1"><i><font color="Green" title="E110">A122</font></i></a>, <a class="ref" href="xboole_1.html#T50" title="XBOOLE_1:th.50">XBOOLE_1:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_16"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c44">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_16"/><i><font color="Green" title="E113">A130</font></i>: 
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:83_1_16"/>
( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="#E3:83_1_16"><i><font color="Green" title="E113">A130</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:83_1_16"/><b>then </b>
( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:83_1_16"/>
contradiction
 <b>by </b><i><a class="txt" href="#E35:83_1"><i><font color="Green" title="E79">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E108:83_1"/><b>then </b><i><font color="Green" title="E113">A131</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E109:83_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c41">M2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E18"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/><a NAME="E110:83_1"/><b>then </b>
<font color="Maroon" title="c41">M2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><a NAME="E111:83_1"/><b>then </b><i><font color="Green" title="E114">A132</font></i>: 
<font color="Maroon" title="c41">M2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><div><i><font color="Green" title="E115">A133</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_17"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:83_1_17"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k1o</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E23:83_1"><i><font color="Green" title="E71">A46</font></i></a>, <a class="txt" href="#E26:83_1"><i><font color="Green" title="E72">A48</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k1o</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span><br/><b>let </b><font color="Maroon" title="c44">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k1o</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c44">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> )</span><br/><b>assume </b><a NAME="E2:83_1_17"/>
<font color="Maroon" title="c44">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k1o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c44">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span><br/><a NAME="E3:83_1_17"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c44">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T34" target="_self" title="GLIB_004:th.34">Lm13</a></i>;<br/><b>hence </b><a NAME="E4:83_1_17"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c44">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font>
 <b>by </b><i><a class="txt" href="#E29:83_1"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E116">A134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_18"/><i><font color="Green" title="E117">A135</font></i>: 
<font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_18"/><b>then </b><i><font color="Green" title="E118">A136</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E106:83_1"><i><font color="Green" title="E112">A129</font></i></a>, <a class="ref" href="zfmisc_1.html#T52" title="ZFMISC_1:th.52">ZFMISC_1:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c44">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:83_1_18_1"/>
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span><br/><a NAME="E2:83_1_18_1"/><b>then </b>
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">em</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:83_1_18_1"/>
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="txt" href="#E1:83_1_18"><i><font color="Green" title="E117">A135</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:83_1_18"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E119">A137</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="#E2:83_1_18"><i><font color="Green" title="E118">A136</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c24">Mep</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">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" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="#E43:83_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="#E61:83_1"><i><font color="Green" title="E91">A68</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p5">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="#E108:83_1"><i><font color="Green" title="E113">A131</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>

;<br/><b>consider </b><font color="Maroon" title="c44">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:83_1_18"/><i><font color="Green" title="E120">A138</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c41">M2'</font>,<font color="Maroon" title="c44">k2</font>]
 <b>by </b><i><a class="txt" href="#E6:83_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="#E111:83_1"><i><font color="Green" title="E114">A132</font></i></a></i>;<br/><div><i><font color="Green" title="E121">A139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18_3"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c45">Vr</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c23">V</font>;<br/><b>assume </b><a NAME="E1:83_1_18_3"/><i><font color="Green" title="E122">A140</font></i>: 
<font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_18_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#T30" target="_self" title="GLIB_004:th.30">Lm9</a></i>;<br/><a NAME="E3:83_1_18_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  )
 <b>by </b><i><a class="txt" href="#E1:83_1_18_3"><i><font color="Green" title="E122">A140</font></i></a>, <a class="ref" href="glib_000.html#T34" title="GLIB_000:th.34">GLIB_000:34</a></i>;<br/><a NAME="E4:83_1_18_3"/><b>then </b>
( not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c23">V</font> &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">em</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c23">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E5:83_1_18_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E84:83_1"><i><font color="Green" title="E104">A115</font></i></a>, <a class="ref" href="glib_000.html#D17" title="GLIB_000:def.17">GLIB_000:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c45">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  implies <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font> )</span><br/><b>assume </b><a NAME="E1:83_1_18_4"/><i><font color="Green" title="E122">A141</font></i>: 
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_18_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_18_4_1_1"/>
( <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  or <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="#E34:83_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="txt" href="#E1:83_1_18_4"><i><font color="Green" title="E122">A141</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_18_4_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_18_4_1_1_1"/><i><font color="Green" title="E123">A142</font></i>: 
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span><br/><div class="add"><a NAME="E2:83_1_18_4_1_1_1"/><b>then </b><i><font color="Green" title="E124">A143</font></i>: 
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E29:83_1"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:83_1_18_4_1_1_1"/>
not <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E8:83_1_18"><i><font color="Green" title="E121">A139</font></i></a>, <a class="txt" href="#E1:83_1_18_4_1_1_1"><i><font color="Green" title="E123">A142</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:83_1_18_4_1_1_1"/>
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font>
 <b>by </b><i><a class="txt" href="#E98:83_1"><i><font color="Green" title="E110">A122</font></i></a>, <a class="txt" href="#E2:83_1_18_4_1_1_1"><i><font color="Green" title="E124">A143</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_18_4_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_18_4_1_1_2"/><i><font color="Green" title="E123">A144</font></i>: 
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span><br/><div class="add"><a NAME="E2:83_1_18_4_1_1_2"/><b>then </b><i><font color="Green" title="E124">A145</font></i>: 
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:83_1_18_4_1_1_2"/>
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:83_1_18_4_1_1_2"><i><font color="Green" title="E123">A144</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:83_1_18_4_1_1_2"/><b>then </b>
not <font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E83:83_1"><i><font color="Green" title="E103">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:83_1_18_4_1_1_2"/>
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font>
 <b>by </b><i><a class="txt" href="#E98:83_1"><i><font color="Green" title="E110">A122</font></i></a>, <a class="txt" href="#E2:83_1_18_4_1_1_2"><i><font color="Green" title="E124">A145</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:83_1_18_4"/>
<font color="Maroon" title="c45">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:83_1_18"/><b>then </b><i><font color="Green" title="E122">A146</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green" title="E123">A147</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_18_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_18_5"/>
<font color="Maroon" title="c44">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:83_1_18_5"/><b>then </b>
<font color="Maroon" title="c44">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:83_1_18_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="ref" href="glib_004.html#T34" target="_self" title="GLIB_004:th.34">Lm13</a></i>;<br/><b>hence </b><a NAME="E4:83_1_18_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:83_1_18"><i><font color="Green" title="E120">A138</font></i></a>, <a class="txt" href="#E10:83_1_18"><i><font color="Green" title="E122">A146</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:83_1_18"/><i><font color="Green" title="E124">A148</font></i>: 
<font color="Maroon" title="c16">k1o</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">k2</font>
 <b>by </b><i><a class="txt" href="#E14:83_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="#E111:83_1"><i><font color="Green" title="E114">A132</font></i></a>, <a class="txt" href="#E112:83_1"><i><font color="Green" title="E115">A133</font></i></a>, <a class="txt" href="#E5:83_1_18"><i><font color="Green" title="E119">A137</font></i></a>, <a class="txt" href="#E7:83_1_18"><i><font color="Green" title="E120">A138</font></i></a></i>;<br/><a NAME="E13:83_1_18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">k1o</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E26:83_1"><i><font color="Green" title="E72">A48</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>hence </b><a NAME="E14:83_1_18"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:83_1_18"><i><font color="Green" title="E123">A147</font></i></a>, <a class="txt" href="#E12:83_1_18"><i><font color="Green" title="E124">A148</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_19"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_19"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c44">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_19"/><i><font color="Green" title="E117">A149</font></i>: 
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:83_1_19"/>
( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c34">em</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font> )
 <b>by </b><i><a class="txt" href="#E3:83_1_19"><i><font color="Green" title="E117">A149</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:83_1_19"/>
contradiction
 <b>by </b><i><a class="txt" href="#E113:83_1"><i><font color="Green" title="E116">A134</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E115:83_1"/><b>then </b><i><font color="Green" title="E117">A150</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E106:83_1"><i><font color="Green" title="E112">A129</font></i></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_20"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_20"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c44">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:83_1_20"/><i><font color="Green" title="E118">A151</font></i>: 
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:83_1_20"/>
( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="#E3:83_1_20"><i><font color="Green" title="E118">A151</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:83_1_20"/><b>then </b>
( <font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:83_1_20"/>
contradiction
 <b>by </b><i><a class="txt" href="#E35:83_1"><i><font color="Green" title="E79">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E117:83_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>then </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c41">M2'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_004.html#K12" title="GLIB_004:func.12">PRIM:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c16">k1o</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E61:83_1"><i><font color="Green" title="E91">A68</font></i></a>, <a class="txt" href="#E115:83_1"><i><font color="Green" title="E117">A150</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>
;<br/><a NAME="E119:83_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c14">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E14:83_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="txt" href="#E111:83_1"><i><font color="Green" title="E114">A132</font></i></a></i>;<br/><b>hence </b><a NAME="E120:83_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:83"/>
<font color="Maroon" title="c6">G2</font> is   <a href="glib_004.html#NM6" title="GLIB_004:NM.6">minimumSpanningTree</a> of <font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
