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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a> ] <b>means </b> ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K3">RAT</a> <b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  holds <br/> ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<a NAME="E3:6_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">q</font> being  <a href="rat_1.html#NM1">Rational</a> holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> )
 
<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">q</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K3">RAT</a>  <b>by </b><i><a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/>
<a NAME="E2:6_1_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> )
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E4:6_1_1"/>
for <font color="Olive">q</font> being  <a href="rat_1.html#NM1">Rational</a> holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff  ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>


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