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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">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> );<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:79_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E3:79_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E4:79_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:79_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:79_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E7:79_1_1"/>
<font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:79_1_1_1"/><i><font color="Green">E34</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">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_1_1_1"/><i><font color="Green">E35</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">x</font>,<font color="Maroon">y</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/>
<a NAME="E4:79_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<a NAME="E5:79_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E6:79_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:79_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="partfun1.html#E4:79_1_1_1"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="partfun1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:79_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , </i>;<br/>


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