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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></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#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>7</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="funct_1.html#NM1">Function</a>] means (  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 ) ) );<br/>
<a NAME="E3:20"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 );<br/>
<a NAME="E2:20_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:20_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:20_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> 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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:20_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:20_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> 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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E11:20_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120">FINSEQ_2:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E11:20_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E13:20_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E2:20_1"><i><font color="Green">E20</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E7:20_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:20_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:20_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E11:20_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E4:20_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom6.html#E5:20_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E12:20_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:20_1"><i><font color="Green">E24</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:20_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E8:20_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom6.html#E11:20_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E3:20"><i><font color="Green">E18</font></i></a>);<br/></i>
<a NAME="E7:20"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E5:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>5</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>5</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></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">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:20_2"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:20_2"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:20_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E1:20_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:20_2"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:20"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<a NAME="E10:20_2"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<b>thus </b><a NAME="E11:20_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>thus </b><a NAME="E12:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E1:20_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E4:20_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>

<b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E8:20_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E1:20_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E3:20_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E5:20_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E10:20_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E14:20_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E1:20_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>assume </b><a NAME="E15:20_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E17:20_2"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E18:20_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:20_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E6:20_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E8:20_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E14:20_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<a NAME="E19:20_2"/>
<font color="Maroon">c<sub>12</sub></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">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:20_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E5:20_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E8:20_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E14:20_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E15:20_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E20:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E17:20_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:20_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E23:20_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="polynom6.html#E6:20"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom6.html#E7:20_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E15:20_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E24:20_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:20_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E8:20_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E14:20_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E15:20_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E18:20_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E22:20_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E15:20_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E23:20_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="polynom6.html#E24:20_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom6.html#T12" target="_self">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>5</sub></font> 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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E14:20"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E15:20"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:20"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green">E27</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
;<br/>
<a NAME="E18:20"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E19:20"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_4"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></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#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:20_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></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#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E17:20"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E1:20_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:20_4"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E5:20_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:20_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E6:20_4"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:20_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E4:20_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E7:20_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></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">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E5:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E2:20_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E4:20_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:20_4"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E10:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E11:20_4"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E9:20"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E10:20"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:20_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E15:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E16:20_4"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E17:20_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E7:20_4"><i><font color="Green">E35</font></i></a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E2:20_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E4:20_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E15:20"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E5:20_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E9:20_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E11:20_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="polynom6.html#E14:20_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E16:20_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E19:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:20_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E6:20_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:20"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E18:20"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E21:20"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T31">POLYNOM1:31</a></i>;<br/>
<a NAME="E22:20"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></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">divisors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:20"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20_5"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E2:20_5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:20_5"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:20_5"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E7:20_5"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:20_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<a NAME="E8:20_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E21:20"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E1:20_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:20_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:20_5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_5"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E13:20_5"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<a NAME="E14:20_5"/>
(  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>;<br/>
<a NAME="E15:20_5"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E20:20"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom6.html#E5:20_5"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E6:20_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E10:20_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E11:20_5"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom6.html#E12:20_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polynom3.html#T22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <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">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E17:20_5"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E18:20_5"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E19:20_5"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E20:20_5"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E21:20_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom6.html#E5:20_5"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E17:20_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E23:20_5"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E24:20_5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E20:20_5"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E26:20_5"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E27:20_5"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E28:20_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E29:20_5"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E10:20"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom6.html#E16:20"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E30:20_5"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:20_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E11:20_5"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom6.html#E15:20_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom6.html#E26:20_5"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E32:20_5"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E33:20_5"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E29:20_5"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E34:20_5"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom6.html#E18:20_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="polynom6.html#E27:20_5"><i><font color="Green">E51</font></i></a>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>
<a NAME="E35:20_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>12</sub></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">decomp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E19:20_5"><i><font color="Green">E45</font></i></a>, <a class="txt" href="polynom6.html#E21:20_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E36:20_5"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:20_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E32:20_5"><i><font color="Green">E54</font></i></a>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>15</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>18</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E39:20_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E21:20"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E1:20_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then </b><i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E11:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E2:20_5"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E13:20_5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="polynom6.html#E15:20_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E10:20_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E15:20_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom6.html#E26:20_5"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E30:20_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="polynom6.html#E33:20_5"><i><font color="Green">E55</font></i></a>, <a class="txt" href="polynom6.html#E34:20_5"><i><font color="Green">E56</font></i></a>, <a class="txt" href="polynom6.html#E36:20_5"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E41:20_5"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E16</font></i></a>, <a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom6.html#E18:20_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="polynom6.html#E27:20_5"><i><font color="Green">E51</font></i></a>, <a class="txt" href="polynom6.html#E34:20_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E42:20_5"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:20_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E32:20_5"><i><font color="Green">E54</font></i></a>, <a class="txt" href="polynom6.html#E35:20_5"><i><font color="Green">E57</font></i></a>, <a class="txt" href="polynom6.html#E36:20_5"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green">E62</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:20_5"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polynom6.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E41:20_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E42:20_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>
;<br/>
<b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E1:20_5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E7:20_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E4:20_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="polynom6.html#E17:20_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom6.html#E21:20_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polynom6.html#E24:20_5"><i><font color="Green">E49</font></i></a>, <a class="txt" href="polynom6.html#E40:20_5"><i><font color="Green">E59</font></i></a>, <a class="txt" href="polynom6.html#E43:20_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:20"/>
 <a href="polynom1.html#K21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E22:20"><i><font color="Green">E31</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>


</div>
