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

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="int_2.html#NM1" title="INT_2:NM.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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">B</font> = <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<span class="kw">thus </span><a NAME="E1:166"/>
 <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="166_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c8">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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:166_1"/>
<font color="Maroon" title="c8">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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span><br/>

<a NAME="E2:166_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">k</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:166_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T85" target="_self" onmouseover="rs('number08/T85')" onmouseout="rh()">Th85</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">let </span><font color="Maroon" title="c8">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> or <font color="Maroon" title="c8">m</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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:166"/><span class="lab"><font color="Green" title="E125">A1</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</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"> <font color="Maroon" title="c8">m</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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/>
<a NAME="E4:166"/>
<font color="Maroon" title="c9">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:166"><span class="lab"><font color="Green" title="E125">A1</font></span></a>, <a class="ref" href="number08.html#T85" target="_self" onmouseover="rs('number08/T85')" onmouseout="rh()">Th85</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:166"/>
<font color="Maroon" title="c8">m</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="c5">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:166"><span class="lab"><font color="Green" title="E125">A1</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>
