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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>consider </b><font color="Maroon" title="c2">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> <b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  holds <br/> ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<a NAME="E3:6_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> holds <br/> ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">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" title="c3">q</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c3">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  <b>by </b><i><a class="ref" href="rat_1.html#D2" title="RAT_1:def.2">RAT_1:def 2</a></i>;<br/>
<a NAME="E2:6_1_1_1"/>
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E2:6_1_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:6_1_1_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c2">X</font>
;<br/>

<b>thus </b><a NAME="E4:6_1_1"/>
for <font color="Olive" title="b1">q</font> being  <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> holds <br/> ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E3:6_1_1"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>


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