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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">k</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>  implies  <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 not <font color="Maroon" title="c1">k</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font></span><br/>

<a NAME="E2:79"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  holds <br/><span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  implies <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">i</font> as   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span> or  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_1_1_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span><br/><div class="add"><a NAME="E2:79_1_1_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><a NAME="E3:79_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:79_1_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:79_1_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:79_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_3.html#T36" title="NAT_3:th.36">NAT_3:36</a></span>; <a class="txt" 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><a NAME="E4:79_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E5:79_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="moebius1.html#T24" target="_self" title="MOEBIUS1:th.24">Th24</a></span>;<br/><a NAME="E6:79_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_1_1_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">p</font> = 
<font color="Maroon" title="c3">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T5" title="NEWTON:th.5">NEWTON:5</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:79_1_1_1"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></span>; <a class="txt" 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="79_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_1_1_2"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
 not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span><br/><div class="add"><a NAME="E2:79_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></span>;<br/><a NAME="E3:79_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E4:79_1_1_2"/>
 not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1_1_2"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_1_1_2"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1_1_2"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>; <a class="txt" 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><span class="kw">end;</span></div>
<a NAME="E3:79"/><span class="kw">then </span>
 <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:79"/>
 <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" title="NAT_3:th.61">NAT_3:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
