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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>( <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ) <b>-&gt; </b>  <a href="nat_1.html#NM1">Nat</a> = <font color="Maroon">a<sub>1</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) );<br/>
<a NAME="E1:65_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:65_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E3:65_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="newton.html#K13">SetPrimes</a> <b> such that </b><br/><a NAME="E3:65_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="nat_3.html#E1:65_1_1"><i><font color="Green">E35</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<a NAME="E4:65_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<a NAME="E5:65_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>]
 
<b>by </b><i><a class="txt" href="nat_3.html#E3:65_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:65_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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