<?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">x</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:7_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:7_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:7_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:7_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<a NAME="E5:7_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:7_1_2"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">y</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:7_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:7_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>] )
 <b>by </b><i/>;<br/>


</div>
