<?xml version="1.0"?>
<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = $1 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">w</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) );<br/>
<a NAME="E1:66_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">i</font> as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">i</font>)
;<br/>

<b>take </b>
<font color="Maroon" title="c3">i</font>
;<br/>

<b>thus </b><a NAME="E3:66_1_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">i</font>) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> <b> such that </b><br/><a NAME="E3:66_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="nat_3.html#E1:66_1_1"><i><font color="Green" title="E34">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">i</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<a NAME="E4:66_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/>
<a NAME="E5:66_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>]
 
<b>by </b><i><a class="txt" href="nat_3.html#E3:66_1_1"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:66_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font>
 ;<br/>


</div>
