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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<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="E2:146"/><i><font color="Green">E41</font></i>: 
 <a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font> <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="E3:146"/><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> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:146"/>
 <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:146"/><b>then </b><i><font color="Green">E43</font></i>: 
 <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/>;<br/>
<a NAME="E6:146"/>
 <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="E7:146"/><b>then </b><i><font color="Green">E44</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:146_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:146_1"/>
 <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:146_1"/>
<span class="p1"><a href="tarski.html#K4">[</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>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:146"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</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#T13" target="_self">Th13</a>, , <a class="ref" href="polynom1.html#T14" target="_self">Th14</a>, </i>;<br/>

<a NAME="E10:146"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:146_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:146_2"/><b>then </b>
<font color="Maroon">y</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:146_2"/><b>then </b>
<font color="Maroon">y</font> <a href="polynom1.html#R2" target="_self">&lt;='</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:146_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17" target="_self">BagOrder</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:146"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T13" target="_self">Th13</a>, <a class="ref" href="polynom1.html#T14" target="_self">Th14</a>, <a class="ref" href="polynom1.html#T15" target="_self">Th15</a>, </i>;<br/>


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