<?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#NM2" title="GLIB_000:NM.2">_Graph</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">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</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"> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> )</span><br/><span class="kw">set </span><font color="Maroon" title="c5">Sx</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">Tx</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>;<br/><span class="kw">assume </span><a NAME="E1:65_1"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Sx</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">Tx</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_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="E1:65_1_1_1"/>
( <font color="Maroon" title="c7">Sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c8">Tx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="glib_000.html#T28" title="GLIB_000:th.28">GLIB_000:28</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c7">Sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">W</font> being    <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="E3:65_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c7">Sx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:65_1_1_1_1_1"/>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Sx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_1"><span class="lab"><font color="Green" title="E45">A3</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="E2:65_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c8">Tx</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:65_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 not <font color="Maroon" title="c8">Tx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</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/><a NAME="E4:65_1_1_1_1_1"/>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_1"><span class="lab"><font color="Green" title="E45">A3</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="E5:65_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">x</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">Tx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="glib_001.html#T63" title="GLIB_001:th.63">GLIB_001:63</a></span>;<br/><span class="kw">hence </span><a NAME="E6:65_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</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="E5:65_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><a NAME="E6:65_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E1:65_1_1_1_1"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><a NAME="E7:65_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1_1_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:65_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D29" title="GLIB_000:def.29">GLIB_000:def 29</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="65_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c8">Tx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">W</font> being    <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="E3:65_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">Tx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:65_1_1_1_2_1"/>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Tx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_2"><span class="lab"><font color="Green" title="E45">A8</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="E2:65_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c7">Sx</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:65_1_1_1_2_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
 not <font color="Maroon" title="c7">Sx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</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/><a NAME="E4:65_1_1_1_2_1"/>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_2"><span class="lab"><font color="Green" title="E45">A8</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="E5:65_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">x</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c7">Sx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65_1_1_1_2_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="glib_001.html#T63" title="GLIB_001:th.63">GLIB_001:63</a></span>;<br/><span class="kw">hence </span><a NAME="E6:65_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1_1_1_2_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="glib_002.html#D5" target="_self" title="GLIB_002:def.5">Def5</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="E5:65_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><a NAME="E6:65_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E1:65_1_1_1_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><a NAME="E7:65_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1_1_1_2"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:65_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D29" title="GLIB_000:def.29">GLIB_000:def 29</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="E4:65_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</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="E2:65"/><span class="kw">then </span>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T33" title="GLIB_000:th.33">GLIB_000:33</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:65"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
