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

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">n</font> is  <a href="number08.html#V2" title="NUMBER08:attr.2">having_at_least_three_different_prime_divisors</a>  implies <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30 )</span><br/>

<span class="kw">assume </span><a NAME="E1:138"/><span class="lab"><font color="Green" title="E104">A1</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> (  not <font color="Maroon" title="c5">n</font> is  <a href="number08.html#V2" title="NUMBER08:attr.2">having_at_least_three_different_prime_divisors</a>  or <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30 )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font>, <font color="Maroon" title="c8">p3</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><a NAME="E3:138"/><span class="lab"><font color="Green" title="E105">A2</font></span>: 
<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c8">p3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 <span class="kw">and </span><br/><a NAME="E4:138"/><span class="lab"><font color="Green" title="E106">A3</font></span>: 
( <font color="Maroon" title="c6">p1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c8">p3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D8" target="_self" onmouseover="rs('number08/D8')" onmouseout="rh()">NUMBER08:def 8</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="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30</span><br/>

<a NAME="E5:138"/>
( ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 ) or ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 ) or ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 ) or ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 ) or ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 ) or ( <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 5 &amp; <font color="Maroon" title="c7">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="number08.html#T49" target="_self" onmouseover="rs('number08/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E6:138"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A4</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 5 or <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 5</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2 or <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="number08.html#E69"><span class="lab"><font color="Green" title="E59">Lm15</font></span></a></span>;<br/>
<a NAME="E7:138"/><span class="lab"><font color="Green" title="E108">A5</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="txt" href="#E4:138"><span class="lab"><font color="Green" title="E106">A3</font></span></a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>;<br/>
<a NAME="E8:138"/>
( <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c8">p3</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c8">p3</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>;<br/>
<a NAME="E9:138"/><span class="kw">then </span>
<font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c8">p3</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T14" onmouseover="rs('euler_1/T14')" onmouseout="rh()">EULER_1:14</a></span>;<br/>
<a NAME="E10:138"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">p3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="txt" href="#E4:138"><span class="lab"><font color="Green" title="E106">A3</font></span></a>, <a class="txt" href="#E7:138"><span class="lab"><font color="Green" title="E108">A5</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a>, <a class="ref" href="pepin.html#T4" onmouseover="rs('pepin/T4')" onmouseout="rh()">PEPIN:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:138"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 30
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:138"><span class="lab"><font color="Green" title="E107">A4</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>
