<?xml version="1.0"?>
<div class="add">

<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_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</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="E2:9_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">x</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>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:9_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:9_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:9_1_1_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:9_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

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

<b>thus </b><a NAME="E5:9_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies  ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><a NAME="E6:9_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:9_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E8:9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , </i>;<br/>


</div>
