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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font>, <font color="Olive" title="b3">Y1</font>, <font color="Olive" title="b4">Y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">X2</font> &amp; <font color="Olive" title="b3">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Y2</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b3">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Olive" title="b2">X2</font>,<font color="Olive" title="b4">Y2</font></span><br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font>, <font color="Maroon" title="c4">Y1</font>, <font color="Maroon" title="c5">Y2</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="c2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</font> implies <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font> )</span><br/>



<b>assume </b><a NAME="E1:135"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X2</font> &amp; <font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y2</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">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font> implies <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font> )</span><br/><b>assume </b><a NAME="E1:135_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font></span><br/><a NAME="E2:135_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def32</a></i>;<br/><a NAME="E3:135_1"/><b>then </b>
( <font color="Maroon" title="c6">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="c1">G</font> &amp; ( ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y1</font> ) or ( <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="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X1</font> ) ) )
 <b>by </b><i><a class="ref" href="glib_000.html#D17" target="_self" title="GLIB_000:def.17">Def17</a></i>;<br/><a NAME="E4:135_1"/><b>then </b>
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E1:135"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="glib_000.html#D17" target="_self" title="GLIB_000:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E5:135_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:135"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c2">X1</font>,<font color="Maroon" title="c4">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesBetween</a> <font color="Maroon" title="c3">X2</font>,<font color="Maroon" title="c5">Y2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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