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

<b>let </b><font color="Maroon">a</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:80"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:80"/>
 <a href="polynom2.html#K1">support</a> <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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i/>;<br/>
<a NAME="E3:80_1"/><i><font color="Green">E47</font></i>: 
<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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E4:80_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:80_1"/>
<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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:80_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="txt" href="nat_3.html#E3:80_1"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E7:80_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:80_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:80"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:80"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:80"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E6:80"/><b>then </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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:80"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <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">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


</div>
