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<b>let </b><font color="Maroon">GS</font> be    <a href="glib_000.html#M1" target="_self">GraphStruct</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>let </b><font color="Maroon">n1</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n2</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:114"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n2</font>
 ;<br/>

<b>set </b><font color="Maroon">G2</font> = <font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">G3</font> = <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E2:114"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E3:114"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">GS</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E4:114"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:114"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#D21" target="_self">Def21</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:114"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D20" target="_self">Def20</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<a NAME="E7:114"/>
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:114"/>
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D20" target="_self">Def20</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
<b>by </b><i><a class="ref" href="glib_000.html#D19" target="_self">Def19</a>, <a class="txt" href="glib_000.html#E22"><i><font color="Green">Lemma50</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="glib_000.html#D18" target="_self">Def18</a></i>
;<br/>

<b>thus </b><a NAME="E10:114"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">GS</font> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n1</font>,<font color="Maroon">x</font></span>)</span> <a href="glib_000.html#K13" target="_self">.set</a> <font color="Maroon">n2</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D18" target="_self">Def18</a></i>;<br/>


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