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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">p</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="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</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">  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span> </span><br/>

<a NAME="E2:21"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is    <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>  : <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span>  )</span><br/>

<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E3:21_1_1"/>
<font color="Maroon" title="c4">x9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="moebius1.html#T39" onmouseover="rs('moebius1/T39')" onmouseout="rh()">MOEBIUS1:39</a></span>;<br/><a NAME="E4:21_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</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 <br/>( <font color="Maroon" title="c4">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="pepin.html#T34" onmouseover="rs('pepin/T34')" onmouseout="rh()">PEPIN:34</a></span>;<br/><span class="kw">hence </span><a NAME="E5:21_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </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="E2:21_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E3:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 ex <font color="Olive" title="b1">t</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 <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E5:21_1"/>
<font color="Maroon" title="c4">x9</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E24">A3</font></span></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">hence </span><a NAME="E3:21"/>
 <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI: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>
