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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<a NAME="E1:156"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">E</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:156"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">E</font></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:156_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">E</font></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span>
 ;<br/>

<a NAME="E2:156_1"/><b>then </b>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">E</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:156_1"/><b>then </b>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:156_1"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">e</font> = <span class="p1"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">E</font></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:156"/><i><font color="Green" title="E87">A1</font></i>: 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T71" target="_self" title="POLYNOM1:th.71">Th71</a></i>;<br/>
<a NAME="E6:156"/><i><font color="Green" title="E88">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E7:156"/><b>then </b><i><font color="Green" title="E89">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:156"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E8:156"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c3">e</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:156_2"/><i><font color="Green" title="E90">A4</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">e</font>
 <b>and </b><br/><a NAME="E2:156_2"/><i><font color="Green" title="E91">A5</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<a NAME="E3:156_2"/><i><font color="Green" title="E92">A6</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="polynom1.html#E6:156"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="polynom1.html#E1:156_2"><i><font color="Green" title="E90">A4</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:156_2"/><b>then </b><i><font color="Green" title="E93">A7</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <span class="p2">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:156"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">e</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c2">E</font>
<b>by </b><i><a class="txt" href="polynom1.html#E3:156_2"><i><font color="Green" title="E92">A6</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c5">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom1.html#E2:156_2"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="polynom1.html#E4:156_2"><i><font color="Green" title="E93">A7</font></i></a>, <a class="ref" href="polynom1.html#T58" target="_self" title="POLYNOM1:th.58">Th58</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:156"/>
 <a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrlin.html#K3" title="MATRLIN:func.3">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><a href="matrlin.html#K3" title="MATRLIN:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom1.html#E7:156"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>


</div>
