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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a>  <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">src</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> (  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Olive" title="b1">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Olive" title="b1">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Olive" title="b1">src</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">src</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> holds <br/> (  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">DCS</font> =  <a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">RFS</font> = <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Gn1a</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">BestEdges</font> =  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">SGn</font> =  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">TGn</font> =  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font> implies  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:44_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( <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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> &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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> );<br/><span class="kw">assume </span><a NAME="E1:44_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c11">BE1</font> being   <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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:44_1_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">BE1</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S1" title="SUBSET_1:sch.1">SUBSET_1:sch 1</a>();<br/></span><a NAME="E4:44_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T19" target="_self" title="GLIB_004:th.19">Th19</a></span>;<br/><a NAME="E5:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:44_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K8" title="GLIB_004:func.8">DIJK:Init</a> <font color="Maroon" title="c2">src</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D11" target="_self" title="GLIB_004:def.11">Def11</a></span>;<br/><a NAME="E2:44_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">src</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T17" target="_self" title="GLIB_004:th.17">Th17</a></span>;<br/><a NAME="E3:44_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c2">src</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E4:44_1_1_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c11">BE1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c12">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:44_1_1_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <span class="kw">and </span><br/><a NAME="E7:44_1_1_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 not <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="xboole_0.html#T6" title="XBOOLE_0:th.6">XBOOLE_0:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">v</font> = <font color="Maroon" title="c12">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44_1_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">W</font> being   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E10:44_1_1_1"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c14">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">src</font>,<font color="Maroon" title="c13">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44_1_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="glib_002.html#D6" title="GLIB_002:def.6">GLIB_002:def 6</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>(  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="c14">W</font> &amp;  not <font color="Maroon" title="c14">W</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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> );<br/><font color="Maroon" title="c14">W</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="c14">W</font></span>)</span> = 
<font color="Maroon" title="c14">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:44_1_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/><a NAME="E12:44_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 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>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44_1_1_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:44_1_1_1"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">k</font>] &amp; (  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</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">m</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E12:44_1_1_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>);<br/></span><a NAME="E15:44_1_1_1"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T18" target="_self" title="GLIB_004:th.18">Th18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:44_1_1_1_2_1"/>
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_2_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> = 
<font color="Maroon" title="c14">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c2">src</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:44_1_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:44_1_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E15:44_1_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c16">k9</font> = <font color="Maroon" title="c15">k</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E3:44_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E4:44_1_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:44_1_1_1_2_1_2"/><span class="kw">then </span>
1 <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> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:44_1_1_1_2_1_2"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">k2a</font> = <font color="Maroon" title="c16">k9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">e</font> = <font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E8:44_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c14">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E9:44_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k2a</font>,<font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T122" title="GLIB_001:th.122">GLIB_001:122</a></span>;<br/><a NAME="E10:44_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>;<br/><a NAME="E11:44_1_1_1_2_1_2"/>
<font color="Maroon" title="c17">k2a</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E12:44_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k2a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E8:44_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a></span>;<br/><a NAME="E13:44_1_1_1_2_1_2"/>
( <font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k2a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k2a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E14:44_1_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:44_1_1_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E14:44_1_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E12:44_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E10:44_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:44_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">BE1</font> = <font color="Maroon" title="c11">BE1</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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">BE1</font>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">e1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">BE1</font><span class="kw"> such that </span><br/><a NAME="E9:44_1_1"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
 for <font color="Olive" title="b1">e2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">BE1</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">e1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e2</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></span><a NAME="E10:44_1_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><a NAME="E11:44_1_1"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c13">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><a NAME="E12:44_1_1"/><span class="kw">then </span>
<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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</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">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E13:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:44_1_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E38">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">y</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>,<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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span>,<font color="Maroon" title="c1">G</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">e1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:44_1_1_2"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>,<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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">e1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><br/><a NAME="E2:44_1_1_2"/><span class="kw">then </span>
<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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></span>;<br/><a NAME="E3:44_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A24</font></span>: 
 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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:44_1_1_2"/>
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></span>;<br/><a NAME="E5:44_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">BE1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E3:44_1_1_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:44_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">e1</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></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">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:44_1_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:44_1_1"/>
<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">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44_1_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/><a NAME="E16:44_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">e1</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span>,<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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:44_1_1"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E13:44_1_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></span>;<br/><span class="kw">hence </span><a NAME="E17:44_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E14:44_1_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="glib_004.html#D7" target="_self" title="GLIB_004:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:44_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:44"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E3:44"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K7" title="GLIB_004:func.7">DIJK:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#D11" target="_self" title="GLIB_004:def.11">Def11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_2"/>
 <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:44_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A27</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="glib_004.html#T15" target="_self" title="GLIB_004:th.15">Th15</a></span>;<br/><a NAME="E3:44_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a></span>)</span></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T19" target="_self" title="GLIB_004:th.19">Th19</a>, <a class="ref" href="card_1.html#T11" title="CARD_1:th.11">CARD_1:11</a></span>;<br/><a NAME="E4:44_2"/><span class="kw">then </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"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></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="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:44_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:44"/>
 <a href="glib_004.html#K6" title="GLIB_004:func.6">DIJK:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K9" title="GLIB_004:func.9">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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