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

<b>thus </b><a NAME="E1:6_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K3">RAT</a>  implies 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">R</font> iff  ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n</font> ) )
 <b>by </b><i>, </i>;<br/>

<b>assume </b><a NAME="E2:6_1_1"/><i><font color="Green">E28</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">R</font> iff  ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n</font> )
 ;<br/>

<b>thus </b><a NAME="E3:6_1_1"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K3">RAT</a> 
  <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="6_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="E1:6_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:6_1_1_1"/><b>then </b>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 <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="E4:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 ;<br/>

<a NAME="E5:6_1_1"/><b>then </b>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div>
