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

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>consider </b><font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E2:148"/><i><font color="Green">E41</font></i>: 
 <a href="polynom1.html#K20" target="_self">divisors</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:148"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3" target="_self">divides</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:148"/>
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:148"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E6:148"/>
 <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E7:148"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:148"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_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="E1:148_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:148_1"/>
<font color="Maroon">b</font> <a href="polynom1.html#R3" target="_self">divides</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T14" target="_self">Th14</a>, <a class="ref" href="polynom1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E4:148_1"/><b>then </b>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:148_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"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></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>
<a NAME="E9:148"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:148"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T15" target="_self">Th15</a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <span class="p3">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i><a class="ref" href="polynom1.html#T17" target="_self">Th17</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E12:148"/>
 <a href="polynom1.html#K20" target="_self">divisors</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>


</div>
