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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> holds <br/>( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> implies ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:186"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">hence </span><a NAME="E2:186"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:186"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<a NAME="E4:186"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></span>;<br/>
<a NAME="E5:186"/><span class="lab"><font color="Green" title="E76">A3</font></span>: 
 <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E77">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="186_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">e</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> )</span><br/><a NAME="E1:186_1"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Maroon" title="c5">e</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="c2">G2</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:186"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E5:186"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="ref" href="glib_000.html#D28" target="_self" title="GLIB_000:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E2:186_1"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D28" target="_self" title="GLIB_000:def.28">Def28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:186"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<a NAME="E8:186"/><span class="lab"><font color="Green" title="E79">A6</font></span>: 
 <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E80">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="186_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">e</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> )</span><br/><a NAME="E1:186_2"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Maroon" title="c5">e</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="c2">G2</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="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:186"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="#E8:186"><span class="lab"><font color="Green" title="E79">A6</font></span></a>, <a class="ref" href="glib_000.html#D29" target="_self" title="GLIB_000:def.29">Def29</a></span>;<br/><span class="kw">hence </span><a NAME="E2:186_2"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D29" target="_self" title="GLIB_000:def.29">Def29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:186"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E11:186"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:186"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E9:186"><span class="lab"><font color="Green" title="E80">A7</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E12:186"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:186"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="txt" href="#E10:186"><span class="lab"><font color="Green" title="E81">A8</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="186_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">e</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/><a NAME="E1:186_3"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></span>;<br/><a NAME="E2:186_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c2">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="ref" href="glib_000.html#T91" target="_self" title="GLIB_000:th.91">Th91</a></span>;<br/><span class="kw">hence </span><a NAME="E3:186_3"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:186"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
