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<b>let </b><font color="Maroon">A1</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">A2</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 <b>and </b><br/><a NAME="E2:5_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] )
 ;<br/>

<b>thus </b><a NAME="E3:5_1"/>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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="E1:5_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:5_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:5_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<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="E4:5_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E5:5_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E7:5_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


</div>
