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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4001 implies  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> ) )</span><br/>

<a NAME="E1:47"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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="nat_1.html#K2" title="NAT_1:func.2">*</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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b2">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:47_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E3:47_1"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47_1"/>
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:47"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4001 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4001 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E46">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><span class="kw">suppose </span></a><a NAME="E1:47_2_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 ;<br/><a NAME="E3:47_2_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_1"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="int_2.html#T28" onmouseover="rs('int_2/T28')" onmouseout="rh()">INT_2:28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2"><span class="kw">suppose </span></a><a NAME="E1:47_2_2"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 2
 ;<br/><a NAME="E3:47_2_2"/><span class="kw">then </span>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_2"><span class="lab"><font color="Green" title="E47">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_2"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_2"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="pepin.html#T41" onmouseover="rs('pepin/T41')" onmouseout="rh()">PEPIN:41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_3"><span class="kw">suppose </span></a><a NAME="E1:47_2_3"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 3
 ;<br/><a NAME="E3:47_2_3"/><span class="kw">then </span>
5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_3"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_3"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_3"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="pepin.html#T59" onmouseover="rs('pepin/T59')" onmouseout="rh()">PEPIN:59</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_4"><span class="kw">suppose </span></a><a NAME="E1:47_2_4"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 7 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 5
 ;<br/><a NAME="E3:47_2_4"/><span class="kw">then </span>
7 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_4"><span class="lab"><font color="Green" title="E47">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_4"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_4"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="ref" href="nat_4.html#T26" target="_self" onmouseover="rs('nat_4/T26')" onmouseout="rh()">Th26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_5"><span class="kw">suppose </span></a><a NAME="E1:47_2_5"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 13 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 7
 ;<br/><a NAME="E3:47_2_5"/><span class="kw">then </span>
13 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_5"><span class="lab"><font color="Green" title="E47">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_5"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_5"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="nat_4.html#T28" target="_self" onmouseover="rs('nat_4/T28')" onmouseout="rh()">Th28</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_6"><span class="kw">suppose </span></a><a NAME="E1:47_2_6"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 23 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 13
 ;<br/><a NAME="E3:47_2_6"/><span class="kw">then </span>
23 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_6"><span class="lab"><font color="Green" title="E47">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_6"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_6"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="nat_4.html#T30" target="_self" onmouseover="rs('nat_4/T30')" onmouseout="rh()">Th30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_7"><span class="kw">suppose </span></a><a NAME="E1:47_2_7"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 43 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 23
 ;<br/><a NAME="E3:47_2_7"/><span class="kw">then </span>
43 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_7"><span class="lab"><font color="Green" title="E47">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_7"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_7"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="nat_4.html#T32" target="_self" onmouseover="rs('nat_4/T32')" onmouseout="rh()">Th32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_8"><span class="kw">suppose </span></a><a NAME="E1:47_2_8"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 83 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 43
 ;<br/><a NAME="E3:47_2_8"/><span class="kw">then </span>
83 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_8"><span class="lab"><font color="Green" title="E47">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_8"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_8"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="nat_4.html#T33" target="_self" onmouseover="rs('nat_4/T33')" onmouseout="rh()">Th33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_9"><span class="kw">suppose </span></a><a NAME="E1:47_2_9"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 163 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 83
 ;<br/><a NAME="E3:47_2_9"/><span class="kw">then </span>
163 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_9"><span class="lab"><font color="Green" title="E47">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_9"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_9"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="nat_4.html#T35" target="_self" onmouseover="rs('nat_4/T35')" onmouseout="rh()">Th35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_10"><span class="kw">suppose </span></a><a NAME="E1:47_2_10"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 317 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 163
 ;<br/><a NAME="E3:47_2_10"/><span class="kw">then </span>
317 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_10"><span class="lab"><font color="Green" title="E47">A14</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_10"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_10"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="nat_4.html#T36" target="_self" onmouseover="rs('nat_4/T36')" onmouseout="rh()">Th36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_11"><span class="kw">suppose </span></a><a NAME="E1:47_2_11"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 631 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 317
 ;<br/><a NAME="E3:47_2_11"/><span class="kw">then </span>
631 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_11"><span class="lab"><font color="Green" title="E47">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_11"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_11"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="nat_4.html#T37" target="_self" onmouseover="rs('nat_4/T37')" onmouseout="rh()">Th37</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_12"><span class="kw">suppose </span></a><a NAME="E1:47_2_12"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 1259 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 631
 ;<br/><a NAME="E3:47_2_12"/><span class="kw">then </span>
1259 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_12"><span class="lab"><font color="Green" title="E47">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_12"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_12"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="nat_4.html#T38" target="_self" onmouseover="rs('nat_4/T38')" onmouseout="rh()">Th38</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_13"><span class="kw">suppose </span></a><a NAME="E1:47_2_13"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 2503 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 1259
 ;<br/><a NAME="E3:47_2_13"/><span class="kw">then </span>
2503 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_13"><span class="lab"><font color="Green" title="E47">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_13"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_13"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="ref" href="nat_4.html#T39" target="_self" onmouseover="rs('nat_4/T39')" onmouseout="rh()">Th39</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_14"><span class="kw">suppose </span></a><a NAME="E1:47_2_14"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
( 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#NR3" title="XXREAL_0:NR.3">&lt;</a> 4001 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )</span><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="nat_1.html#K2" title="NAT_1:func.2">*</a> 2503
 ;<br/><a NAME="E3:47_2_14"/><span class="kw">then </span>
4001 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:47_2_14"><span class="lab"><font color="Green" title="E47">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_2_14"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2_14"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="nat_4.html#T40" target="_self" onmouseover="rs('nat_4/T40')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
