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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#M2" target="_self">Subgraph</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_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/><b>assume </b><a NAME="E1:169_1"/><i><font color="Green">E28</font></i>: 
<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>
 ;<br/><a NAME="E2:169_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26" 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/>;<br/><a NAME="E3:169_1"/>
<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#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:169_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>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:169"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</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#D3">TARSKI:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_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/><b>assume </b><a NAME="E1:169_2"/><i><font color="Green">E41</font></i>: 
<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>
 ;<br/><a NAME="E2:169_2"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26" 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/>;<br/><a NAME="E3:169_2"/>
<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#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:169_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>
 <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="E4:169"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</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#D3">TARSKI:def 3</a></i>;<br/>

<a NAME="E5:169"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E6:169"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E7:169"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T10">XBOOLE_1:10</a></i>;<br/>
<a NAME="E8:169"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E9:169"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D29" target="_self">Def29</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E10:169"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K21" target="_self">.edgesInOut</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E11:169"/>
<span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E12:169"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K19" target="_self">.edgesInto</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K20" target="_self">.edgesOutOf</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E13:169"/>
<font color="Maroon">G2</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22" target="_self">.edgesBetween</a> <font color="Maroon">X</font>
 ;<br/>


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