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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</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">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</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">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> ) )</span><br/>







<b>assume </b><a NAME="E1:182"/><i><font color="Green" title="E75">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</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">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<b>hence </b><a NAME="E2:182"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K15" title="GLIB_000:func.15">.order()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</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">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<b>thus </b><a NAME="E3:182"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> 
 <b>by </b><i><a class="txt" href="glib_000.html#E1:182"><i><font color="Green" title="E75">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</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">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<a NAME="E4:182"/><i><font color="Green" title="E76">A2</font></i>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> &amp;  <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> &amp;  <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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="glib_000.html#E1:182"><i><font color="Green" title="E75">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self" title="GLIB_000:def.36">Def36</a></i>;<br/>
<div><i><font color="Green" title="E77">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="182_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">.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#K19" title="GLIB_000:func.19">.edgesInto</a> <font color="Maroon" title="c3">X</font> )</span><br/><a NAME="E1:182_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#K19" title="GLIB_000:func.19">.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> ) )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:182"><i><font color="Green" title="E76">A2</font></i></a>, <a class="ref" href="glib_000.html#D28" target="_self" title="GLIB_000:def.28">Def28</a></i>;<br/><b>hence </b><a NAME="E2:182_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#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self" title="GLIB_000:def.28">Def28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:182"/><i><font color="Green" title="E78">A4</font></i>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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#K19" title="GLIB_000:func.19">.edgesInto</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.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#K20" title="GLIB_000:func.20">.edgesOutOf</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">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<div><i><font color="Green" title="E79">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K20" title="GLIB_000:func.20">.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#K20" title="GLIB_000:func.20">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> )</span><br/><a NAME="E1:182_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#K20" title="GLIB_000:func.20">.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> ) )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:182"><i><font color="Green" title="E76">A2</font></i></a>, <a class="ref" href="glib_000.html#D29" target="_self" title="GLIB_000:def.29">Def29</a></i>;<br/><b>hence </b><a NAME="E2:182_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#K20" title="GLIB_000:func.20">.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#K20" title="GLIB_000:func.20">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D29" target="_self" title="GLIB_000:def.29">Def29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:182"/><i><font color="Green" title="E80">A6</font></i>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.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#K20" title="GLIB_000:func.20">.edgesOutOf</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<b>thus </b><a NAME="E9:182"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.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#K21" title="GLIB_000:func.21">.edgesInOut</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E6:182"><i><font color="Green" title="E78">A4</font></i></a>, <a class="txt" href="glib_000.html#E7:182"><i><font color="Green" title="E79">A5</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>

<b>thus </b><a NAME="E10:182"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E5:182"><i><font color="Green" title="E77">A3</font></i></a>, <a class="txt" href="glib_000.html#E8:182"><i><font color="Green" title="E80">A6</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.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="182_3"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/><a NAME="E1:182_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#K24" title="GLIB_000:func.24">.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> )
 <b>by </b><i><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></i>;<br/><a NAME="E2:182_3"/><b>then </b>
( <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#K24" title="GLIB_000:func.24">.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> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:182"><i><font color="Green" title="E75">A1</font></i></a>, <a class="ref" href="glib_000.html#T91" target="_self" title="GLIB_000:th.91">Th91</a></i>;<br/><b>hence </b><a NAME="E3:182_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#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:182"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.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#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
