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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:183"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">G2</font>
 ;<br/>

<b>thus </b><a NAME="E2:183"/>
<font color="Maroon">G1</font> <a href="glib_000.html#K15" target="_self">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K15" target="_self">.order()</a> 
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E3:183"/>
<font color="Maroon">G1</font> <a href="glib_000.html#K17" target="_self">.size()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K17" target="_self">.size()</a> 
 <b>by </b><i>, </i>;<br/>

<a NAME="E4:183"/><i><font color="Green">E29</font></i>: 
(  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> iff ( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:183_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> iff <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:183"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> iff ( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:183_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> iff <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:183"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>hence </b><a NAME="E9:183"/>
<font color="Maroon">G1</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:183"/>
<font color="Maroon">G1</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_3"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> iff <font color="Maroon">e</font> <a href="glib_000.html#R4" target="_self">DSJoins</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">G1</font> )
 <b>by </b><i/>;<br/><a NAME="E2:183_3"/><b>then </b>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> iff <font color="Maroon">e</font> <a href="glib_000.html#R4" target="_self">DSJoins</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">G2</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:183_3"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G2</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> iff <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:183"/>
<font color="Maroon">G1</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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