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

<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:90"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">i</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">a</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D6">POLYNOM1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:90"/>
 <a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
