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<b>let </b><font color="Maroon" title="c1">GS</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">GraphStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">n2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:113"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c4">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">G2</font> = <font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font>;<br/>
<b>set </b><font color="Maroon" title="c7">G3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font>;<br/>
<a NAME="E2:113"/><i><font color="Green" title="E50">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">n2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#T9" target="_self" title="GLIB_000:th.9">Th9</a></i>;<br/>
<a NAME="E3:113"/><i><font color="Green" title="E51">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#T9" target="_self" title="GLIB_000:th.9">Th9</a></i>;<br/>
<a NAME="E4:113"/>
<font color="Maroon" title="c4">n1</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="c4">n1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:113"/><b>then </b>
<font color="Maroon" title="c4">n1</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="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_000.html#E3:113"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:113"/>
<font color="Maroon" title="c4">n1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:113"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>

<a NAME="E7:113"/>
<font color="Maroon" title="c5">n2</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="c5">n2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:113"/>
<font color="Maroon" title="c5">n2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:113"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n1</font>
<b>by </b><i><a class="txt" href="glib_000.html#E1:113"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="glib_000.html#T12" target="_self" title="GLIB_000:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font>
<b>by </b><i><a class="ref" href="glib_000.html#T11" target="_self" title="GLIB_000:th.11">Th11</a></i>
;<br/>

<b>thus </b><a NAME="E10:113"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">GS</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <font color="Maroon" title="c5">n2</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T11" target="_self" title="GLIB_000:th.11">Th11</a></i>;<br/>


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