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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:183"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K15" target="_self">.order()</a>  = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_000.html#E1:183"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self">Def36</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K15" target="_self">.order()</a> 

;<br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K17" target="_self">.size()</a>  = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_000.html#E1:183"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self">Def36</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K17" target="_self">.size()</a> 

;<br/>

<a NAME="E4:183"/><i><font color="Green">E78</font></i>: 
(  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="glib_000.html#E1:183"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_000.html#D36" target="_self">Def36</a></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:183"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#D28" target="_self">Def28</a></i>;<br/><b>hence </b><a NAME="E2:183_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self">Def28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:183"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></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_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="glib_000.html#E4:183"><i><font color="Green">E78</font></i></a>, <a class="ref" href="glib_000.html#D29" target="_self">Def29</a></i>;<br/><b>hence </b><a NAME="E2:183_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D29" target="_self">Def29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:183"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_000.html#E6:183"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">c<sub>3</sub></font>

;<br/><br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_000.html#E6:183"><i><font color="Green">E79</font></i></a>, <a class="txt" href="glib_000.html#E8:183"><i><font color="Green">E80</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">c<sub>3</sub></font>

;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:183_7"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R4" target="_self">DSJoins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D33" target="_self">Def33</a></i>;<br/><a NAME="E2:183_7"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="glib_000.html#R4" target="_self">DSJoins</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:183"><i><font color="Green">E77</font></i></a>, <a class="ref" href="glib_000.html#T91" target="_self">Th91</a></i>;<br/><b>hence </b><a NAME="E3:183_7"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D33" target="_self">Def33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:183"/>
<font color="Maroon">c<sub>1</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="glib_000.html#K24" target="_self">.edgesDBetween</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
