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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">r</font></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:1_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> iff ( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> iff ( <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">o</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:1_1_1"/>
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">r</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">C'</font> = <font color="Maroon">C</font> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:1_1_1"/>
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/> ex <font color="Olive">p</font>, <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">r</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">C'</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">D'</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E3:1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</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">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:1_1_1_1"/><i><font color="Green">E64</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/><b>take </b><font color="Maroon">q</font> = <font color="Maroon">q</font>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>take </b><font color="Maroon">s</font> = <font color="Maroon">s</font>;<br/><b>thus </b><a NAME="E6:1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E7:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E8:1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>given </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:1_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">s</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">q</font>,<font color="Maroon">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E2:1_1_1_2"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:1_1_1"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff  ex <font color="Olive">q</font>, <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">q</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">q</font>,<font color="Olive">r</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 <b>by </b><i/>;<br/>


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