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

<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E46">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4001 )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E47">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">p</font> being   <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>   st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">p</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E48">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 ) or ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 3 ) or ( 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 5 ) or ( 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 7 ) or ( 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 13 ) or ( 13 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 23 ) or ( 23 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 43 ) or ( 43 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 83 ) or ( 83 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 163 ) or ( 163 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 317 ) or ( 317 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 631 ) or ( 631 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1259 ) or ( 1259 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2503 ) or ( 2503 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4001 ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E1:47"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><b>suppose </b></a><a NAME="E1:47_2_1"/><i><font color="Green" title="E48">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 )
 ;<br/><div class="add"><a NAME="E2:47_2_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 ;<br/><a NAME="E3:47_2_1"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2"><b>suppose </b></a><a NAME="E1:47_2_2"/><i><font color="Green" title="E48">A5</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 3 )
 ;<br/><div class="add"><a NAME="E2:47_2_2"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2
 ;<br/><a NAME="E3:47_2_2"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 3 &amp; 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="pepin.html#T44" title="PEPIN:th.44">PEPIN:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_3"><b>suppose </b></a><a NAME="E1:47_2_3"/><i><font color="Green" title="E48">A6</font></i>: 
( 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 5 )
 ;<br/><div class="add"><a NAME="E2:47_2_3"/>
5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 3
 ;<br/><a NAME="E3:47_2_3"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 5 &amp; 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_3"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_3"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="pepin.html#T64" title="PEPIN:th.64">PEPIN:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_4"><b>suppose </b></a><a NAME="E1:47_2_4"/><i><font color="Green" title="E48">A7</font></i>: 
( 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 7 )
 ;<br/><div class="add"><a NAME="E2:47_2_4"/>
7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 5
 ;<br/><a NAME="E3:47_2_4"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 7 &amp; 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_4"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_4"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T26" target="_self" title="NAT_4:th.26">Th26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_5"><b>suppose </b></a><a NAME="E1:47_2_5"/><i><font color="Green" title="E48">A8</font></i>: 
( 7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 13 )
 ;<br/><div class="add"><a NAME="E2:47_2_5"/>
13 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 7
 ;<br/><a NAME="E3:47_2_5"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 13 &amp; 13 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_5"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_5"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T28" target="_self" title="NAT_4:th.28">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_6"><b>suppose </b></a><a NAME="E1:47_2_6"/><i><font color="Green" title="E48">A9</font></i>: 
( 13 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 23 )
 ;<br/><div class="add"><a NAME="E2:47_2_6"/>
23 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 13
 ;<br/><a NAME="E3:47_2_6"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 23 &amp; 23 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_6"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_6"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T30" target="_self" title="NAT_4:th.30">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_7"><b>suppose </b></a><a NAME="E1:47_2_7"/><i><font color="Green" title="E48">A10</font></i>: 
( 23 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 43 )
 ;<br/><div class="add"><a NAME="E2:47_2_7"/>
43 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 23
 ;<br/><a NAME="E3:47_2_7"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 43 &amp; 43 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_7"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_7"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T32" target="_self" title="NAT_4:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_8"><b>suppose </b></a><a NAME="E1:47_2_8"/><i><font color="Green" title="E48">A11</font></i>: 
( 43 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 83 )
 ;<br/><div class="add"><a NAME="E2:47_2_8"/>
83 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 43
 ;<br/><a NAME="E3:47_2_8"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 83 &amp; 83 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_8"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_8"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T33" target="_self" title="NAT_4:th.33">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_9"><b>suppose </b></a><a NAME="E1:47_2_9"/><i><font color="Green" title="E48">A12</font></i>: 
( 83 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 163 )
 ;<br/><div class="add"><a NAME="E2:47_2_9"/>
163 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 83
 ;<br/><a NAME="E3:47_2_9"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 163 &amp; 163 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_9"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_9"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T35" target="_self" title="NAT_4:th.35">Th35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_10"><b>suppose </b></a><a NAME="E1:47_2_10"/><i><font color="Green" title="E48">A13</font></i>: 
( 163 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 317 )
 ;<br/><div class="add"><a NAME="E2:47_2_10"/>
317 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 163
 ;<br/><a NAME="E3:47_2_10"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 317 &amp; 317 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_10"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_10"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T36" target="_self" title="NAT_4:th.36">Th36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_11"><b>suppose </b></a><a NAME="E1:47_2_11"/><i><font color="Green" title="E48">A14</font></i>: 
( 317 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 631 )
 ;<br/><div class="add"><a NAME="E2:47_2_11"/>
631 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 317
 ;<br/><a NAME="E3:47_2_11"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 631 &amp; 631 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_11"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_11"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T37" target="_self" title="NAT_4:th.37">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_12"><b>suppose </b></a><a NAME="E1:47_2_12"/><i><font color="Green" title="E48">A15</font></i>: 
( 631 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1259 )
 ;<br/><div class="add"><a NAME="E2:47_2_12"/>
1259 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 631
 ;<br/><a NAME="E3:47_2_12"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1259 &amp; 1259 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_12"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_12"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T38" target="_self" title="NAT_4:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_13"><b>suppose </b></a><a NAME="E1:47_2_13"/><i><font color="Green" title="E48">A16</font></i>: 
( 1259 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2503 )
 ;<br/><div class="add"><a NAME="E2:47_2_13"/>
2503 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1259
 ;<br/><a NAME="E3:47_2_13"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2503 &amp; 2503 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_13"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_13"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T39" target="_self" title="NAT_4:th.39">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_14"><b>suppose </b></a><a NAME="E1:47_2_14"/><i><font color="Green" title="E48">A17</font></i>: 
( 2503 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4001 )
 ;<br/><div class="add"><a NAME="E2:47_2_14"/>
4001 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2503
 ;<br/><a NAME="E3:47_2_14"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 4001 &amp; 4001 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:47_2_14"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="nat_4.html#E2:47"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_14"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T40" target="_self" title="NAT_4:th.40">Th40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
