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<b>let </b><font color="Maroon">P</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K4" target="_self">~</a> 
 ;<br/><a NAME="E2:52_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_1_1"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/><a NAME="E5:52_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:52_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:52_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span> <a href="relat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span> <a href="relat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_1_2"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:52_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:52_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:52_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:52_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K4" target="_self">~</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:52_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K4" target="_self">~</a>  iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span> <a href="relat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:52"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K5" target="_self">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K4" target="_self">~</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span> <a href="relat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relat_1.html#K4" target="_self">~</a> </span>)</span>
 <b>by </b><i/>;<br/>


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