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<b>let </b><font color="Maroon">P</font> be   <a href="relat_1.html#NM1">Relation</a>, <font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K1" target="_self">["</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">R</font></span><a href="yellow_3.html#K1" target="_self">"]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">s</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">q</font>,<font color="Maroon">t</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="E6:21_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:21_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></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/><b>thus </b><a NAME="E8:21_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E9:21_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E10:21_1"/>
 ex <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E11:21_1"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E12:21_1"/>
 ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">e</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:21"/><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green">E29</font></i>: 
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> &amp;  ex <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> &amp;  ex <font color="Olive">e</font>, <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">e</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:21"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:21"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E10:21"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E11:21"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></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">d</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E12:21"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">e</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">d</font>,<font color="Maroon">f</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>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E13:21"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K1" target="_self">["</a><span class="default"><font color="Maroon">P</font>,<font color="Maroon">R</font></span><a href="yellow_3.html#K1" target="_self">"]</a></span>
 <b>by </b><i>, , </i>;<br/>


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