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

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

<a NAME="E1:157"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&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 class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">E</font> = <span class="p1"><a href="finseq_4.html#K2">&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 class="p2">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:157"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">E</font></span><a href="matrlin.html#K3">*&gt;</a></span> <a href="tarski.html#R1">c=</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="157_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:157_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">E</font></span><a href="matrlin.html#K3">*&gt;</a></span>
 ;<br/>

<a NAME="E2:157_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">E</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:157_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:157_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">e</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">E</font></span><a href="matrlin.html#K3">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:157"/><i><font color="Green">E41</font></i>: 
<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> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i/>;<br/>
<a NAME="E6:157"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E7:157"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</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>
 
<b>by </b><i><a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E8:157"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</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="finseq_4.html#K4">/.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">e</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Olive">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="157_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:157_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E2:157_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</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="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:157_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:157_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<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="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <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#D3" target="_self">Def3</a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>
<b>thus </b><font color="Maroon">e</font> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">E</font>
<b>by </b><i><a class="ref" href="polynom1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom1.html#T20" target="_self">Th20</a>, <a class="ref" href="polynom1.html#T22" target="_self">Th22</a>, <a class="ref" href="polynom1.html#T7" target="_self">Th7</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:157"/>
 <a href="polynom1.html#K21" target="_self">decomp</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="matrlin.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="matrlin.html#K3">*&gt;</a></span>
 <b>by </b><i>, </i>;<br/>


</div>
