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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>, <font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</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="E2:151"/><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="E3:151"/><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="E4:151"/>
 <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="E5:151"/><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/>
<b>assume </b><a NAME="E6:151"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom1.html#K9" target="_self">+</a> <font color="Maroon">b2</font>
 ;<br/>

<a NAME="E7:151"/><b>then </b>
<font color="Maroon">b1</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:151"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E9:151"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T16" target="_self">Th16</a>, <a class="ref" href="polynom1.html#T18" target="_self">Th18</a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:151"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E12:151"/><i><font color="Green">E46</font></i>: 
<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> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E13:151"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, </i>;<br/>

<a NAME="E14:151"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">b1</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E15:151"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, </i>;<br/>


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