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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_003.html#NM2" title="GLIB_003:NM.2">EGraph</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">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Olive" title="b2">e2</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_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b3">x</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  holds <br/>( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font> &amp; <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><b>let </b><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font>, <font color="Maroon" title="c4">val</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"> ( not <font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c3">e2</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_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c4">val</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  implies ( <font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font> &amp; <font color="Maroon" title="c2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> ) )</span><br/>



<b>set </b><font color="Maroon" title="c5">Gn</font> = <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c4">val</font>;<br/>
<b>assume </b><a NAME="E1:192"/><i><font color="Green" title="E42">A1</font></i>: 
( not <font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  &amp; <font color="Maroon" title="c3">e2</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_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c4">val</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</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">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font> &amp; <font color="Maroon" title="c2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> )</span><br/>

<a NAME="E2:192"/><b>then </b>
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="glib_003.html#D21" target="_self" title="GLIB_003:def.21">Def21</a></i>;<br/>
<a NAME="E3:192"/><b>then </b>
 <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c4">val</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">val</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_003.html#T39" target="_self" title="GLIB_003:th.39">Th39</a></i>;<br/>
<a NAME="E4:192"/><b>then </b>
( <font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> or <font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e1</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c4">val</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="glib_003.html#E1:192"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></i>;<br/>
<a NAME="E5:192"/><b>then </b>
<font color="Maroon" title="c3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="glib_003.html#E1:192"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:192"/>
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="c2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span><br/>

<b>thus </b><a NAME="E7:192"/>
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E1:192"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="glib_003.html#D21" target="_self" title="GLIB_003:def.21">Def21</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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