<?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  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E1:21"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> 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="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:21_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>hence </b><a NAME="E4:21_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <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="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E1:21"><i><font color="Green">E16</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D26">QUOFIELD:def 26</a><br/>

<b>thus </b><a NAME="E5:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="grcat_1.html#V1">additive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a>,<a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a>,<a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:21_2"/>
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K15">.</a> <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>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> 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="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>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>1</sub></font>, <font color="Maroon">c<sub>23</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>, <font color="Maroon">c<sub>24</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:21_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E4:21_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</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>26</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>, <font color="Maroon">c<sub>27</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:21_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>and </b><br/><a NAME="E8:21_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E9:21_2_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</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>29</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>, <font color="Maroon">c<sub>30</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E11:21_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E12:21_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E13:21_2_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E15:21_2_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="polynom6.html#E8:21_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom6.html#E12:21_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E16:21_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>;<br/>
<a NAME="E17:21_2_1"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E7:21_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E11:21_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="polynom6.html#E15:21_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E18:21_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>;<br/>
<a NAME="E19:21_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E5:21_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom6.html#E9:21_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E13:21_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E15:21_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E18:21_2_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:21"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="group_6.html#V1">multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:21_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E7:21_3_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</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="Olive">b<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</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 <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>3</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>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</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>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> 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="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</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>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="E3:21_3_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:21_3_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E7:21_3_1_1"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</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>23</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:21_3_1_1"/><i><font color="Green">E24</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>24</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#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</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="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="finseq_4.html#K4">/.</a> <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>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<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>, <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>ex <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">a<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>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &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#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>6</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>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <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>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>6</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>7</sub></font>,<font color="Olive">b<sub>8</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>8</sub></font></span>)</span> ) ) );<br/>
<a NAME="E9:21_3_1_1"/><i><font color="Green">E25</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>24</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 the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="finseq_2.html#K3">*</a>  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="21_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1"/>
<font color="Maroon">c<sub>25</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>24</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</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>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green">E23</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>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:21_3_1_1_1"/><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>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>25</sub></font> <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>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>29</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</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>, <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <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>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">a<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>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> );<br/>
<a NAME="E7:21_3_1_1_1"/><i><font color="Green">E27</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">S<sub>3</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="21_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1_1"/>
<font color="Maroon">c<sub>32</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#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>32</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>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:21_3_1_1_1_1"/><i><font color="Green">E28</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>32</sub></font> <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>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>34</sub></font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>35</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>33</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>34</sub></font>
;<br/>

<b>thus </b><a NAME="E6:21_3_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>32</sub></font> <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>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E9:21_3_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>32</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:21_3_1_1_1"/><i><font color="Green">E29</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>32</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#E7:21_3_1_1_1"><i><font color="Green">E27</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <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>33</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>32</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>30</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>31</sub></font>
;<br/>

<b>thus </b><a NAME="E12:21_3_1_1_1"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E13:21_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>25</sub></font> <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>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:21_3_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>32</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>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_3_1_1_1"><i><font color="Green">E28</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>34</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E15:21_3_1_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/>

<b>hence </b><a NAME="E16:21_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <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>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>34</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>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_3_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E11:21_3_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</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>24</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1"/><i><font color="Green">E27</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>24</sub></font></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>25</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#E9:21_3_1_1"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>take </b><font color="Maroon">c<sub>26</sub></font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>25</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</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>29</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>, <font color="Maroon">c<sub>30</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E15:21_3_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E16:21_3_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E17:21_3_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E18:21_3_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E20:21_3_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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>24</sub></font> holds <br/><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<a NAME="E2:21_3_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:21_3_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:21_3_1_1_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>24</sub></font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</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="Maroon">c<sub>22</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E23:21_3_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E24:21_3_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</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>24</sub></font> holds <br/><font color="Maroon">c<sub>33</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="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E26:21_3_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E26:21_3_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E28:21_3_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>35</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_3_1_1_3"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>37</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_3_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E4:21_3_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:21_3_1_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E6:21_3_1_1_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:21_3_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:21_3_1_1_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21_3_1_1_3"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:21_3_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:21_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E12:21_3_1_1_3"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E5:21_3_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_3"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E30:21_3_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>36</sub></font> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</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="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E30:21_3_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
;<br/>
<a NAME="E32:21_3_1_1"/>
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>36</sub></font> holds <br/><font color="Maroon">c<sub>36</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>37</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>38</sub></font>, <font color="Maroon">c<sub>39</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:21_3_1_1_5"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>37</sub></font> <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>38</sub></font>,<font color="Maroon">c<sub>39</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:21_3_1_1_5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:21_3_1_1_5"/>
<font color="Maroon">c<sub>37</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>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green">E37</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>40</sub></font>, <font color="Maroon">c<sub>41</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>42</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>43</sub></font>, <font color="Maroon">c<sub>44</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:21_3_1_1_5"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>and </b><br/><a NAME="E8:21_3_1_1_5"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>37</sub></font>
 <b>and </b><br/><a NAME="E9:21_3_1_1_5"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>37</sub></font> <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>40</sub></font>,<font color="Maroon">c<sub>41</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E10:21_3_1_1_5"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>42</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:21_3_1_1_5"/><i><font color="Green">E44</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>42</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#NM1">bag</a> of <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>22</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="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <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>43</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E12:21_3_1_1_5"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>38</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>41</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_3_1_1_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> = 
<font color="Maroon">c<sub>24</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_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>43</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>44</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_5"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom6.html#E7:21_3_1_1_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1_5"><i><font color="Green">E45</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<b>consider </b><font color="Maroon">c<sub>45</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E15:21_3_1_1_5"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>44</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>45</sub></font>
 <b>and </b><br/><a NAME="E16:21_3_1_1_5"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>45</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_5"/><i><font color="Green">E49</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>45</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#NM1">bag</a> of <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>22</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="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <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>43</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>46</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:21_3_1_1_5_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>46</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>45</sub></font></span>)</span>
 ;<br/><a NAME="E2:21_3_1_1_5_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>47</sub></font>, <font color="Maroon">c<sub>48</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:21_3_1_1_5_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>46</sub></font> <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>47</sub></font>,<font color="Maroon">c<sub>48</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:21_3_1_1_5_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>47</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>48</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E17:21_3_1_1_5"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:21_3_1_1_5_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:21_3_1_1_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5_2"><i><font color="Green">E50</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>49</sub></font>, <font color="Maroon">c<sub>50</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:21_3_1_1_5_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>46</sub></font> <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>49</sub></font>,<font color="Maroon">c<sub>50</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E9:21_3_1_1_5_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>49</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>44</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>50</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1_5"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E10:21_3_1_1_5_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>49</sub></font> &amp; <font color="Maroon">c<sub>48</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>50</sub></font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_5_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_5_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/><a NAME="E11:21_3_1_1_5_2"/>
<font color="Maroon">c<sub>45</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_5_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E12:21_3_1_1_5_2"/>
<font color="Maroon">c<sub>45</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>46</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_3_1_1_5_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="polynom6.html#E6:21_3_1_1_5_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_5_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_5_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:21_3_1_1_5"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:21_3_1_1_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>36</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="polynom6.html#E13:21_3_1_1_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="polynom6.html#E15:21_3_1_1_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polynom6.html#E19:21_3_1_1_5"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>37</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E33:21_3_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrlin.html#K8">Sum</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>37</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E35:21_3_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E36:21_3_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E37:21_3_1_1"/><i><font color="Green">E40</font></i>: 
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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12">RLVECT_1:def 12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>;<br/>
<a NAME="E38:21_3_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">S<sub>3</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="21_3_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>38</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>38</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>39</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>41</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E2:21_3_1_1_6"/>
( <font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>38</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E3:21_3_1_1_6"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>38</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E40:21_3_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E38:21_3_1_1"><i><font color="Green">E41</font></i></a>);<br/></i>
<a NAME="E41:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E42:21_3_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E35:21_3_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E43:21_3_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E36:21_3_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E44:21_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<b>then </b><i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom6.html#E43:21_3_1_1"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
;<br/>
<a NAME="E46:21_3_1_1"/>
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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>39</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>40</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E1:21_3_1_1_8"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:21_3_1_1_8"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>45</sub></font> = <font color="Maroon">c<sub>42</sub></font>, <font color="Maroon">c<sub>46</sub></font> = <font color="Maroon">c<sub>43</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>47</sub></font> = <font color="Maroon">c<sub>44</sub></font>, <font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>45</sub></font>, <font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>46</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E6:21_3_1_1_8"/><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:21_3_1_1_8"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:21_3_1_1_8"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_8"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:21_3_1_1_8"/>
<font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_8"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E7:21_3_1_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:21_3_1_1_8"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>40</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_8"><i><font color="Green">E46</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_8"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E11:21_3_1_1_8"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:21_3_1_1_8"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>47</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E37:21_3_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_8"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>43</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom6.html#E11:21_3_1_1_8"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>41</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1_8"><i><font color="Green">E53</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_8"><i><font color="Green">E51</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E47:21_3_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E42:21_3_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="polynom6.html#E45:21_3_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rlvect_1.html#D12">RLVECT_1:def 12</a></i>;<br/>
<a NAME="E48:21_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E49:21_3_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E15:21_3_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E17:21_3_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E18:21_3_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom6.html#E33:21_3_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="polynom6.html#E47:21_3_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="polynom1.html#T34">POLYNOM1:34</a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</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>, <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 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>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">a<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>1</sub></font>,<font color="Olive">b<sub>2</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>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#K21">decomp</a> <font color="Maroon">c<sub>22</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>, <font color="Olive">b<sub>6</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>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>4</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>5</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Olive">b<sub>4</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>1</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>5</sub></font></span>)</span>,<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><a href="polynom3.html#K2">*&gt;</a></span> ) ) );<br/>
<a NAME="E50:21_3_1_1"/><i><font color="Green">E47</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>24</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">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>  st <font color="Maroon">S<sub>4</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="21_3_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>39</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9"/>
<font color="Maroon">c<sub>39</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>24</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9"/><b>then </b>
<font color="Maroon">c<sub>39</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>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green">E23</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>40</sub></font>, <font color="Maroon">c<sub>41</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:21_3_1_1_9"/><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>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>39</sub></font> <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>40</sub></font>,<font color="Maroon">c<sub>41</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>40</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>40</sub></font>, <font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>41</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>defpred </b><font color="Maroon">S<sub>5</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>, <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#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">a<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>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">a<sub>2</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>42</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> );<br/>
<a NAME="E6:21_3_1_1_9"/><i><font color="Green">E49</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></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 <font color="Maroon">S<sub>5</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="21_3_1_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>44</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9_1"/>
<font color="Maroon">c<sub>44</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#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9_1"/><b>then </b>
<font color="Maroon">c<sub>44</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>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>45</sub></font>, <font color="Maroon">c<sub>46</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:21_3_1_1_9_1"/><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>44</sub></font> <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>45</sub></font>,<font color="Maroon">c<sub>46</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>46</sub></font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>47</sub></font> = <font color="Maroon">c<sub>45</sub></font>, <font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>46</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/>
<b>reconsider </b><font color="Maroon">c<sub>49</sub></font> = <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>42</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>48</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">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>take </b>
<font color="Maroon">c<sub>49</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>47</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>48</sub></font>
;<br/>

<b>thus </b><a NAME="E7:21_3_1_1_9_1"/>
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>44</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>47</sub></font>,<font color="Maroon">c<sub>48</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>49</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>42</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>47</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>48</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_9_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>44</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="E8:21_3_1_1_9"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:21_3_1_1_9"/><i><font color="Green">E51</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"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>44</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#E6:21_3_1_1_9"><i><font color="Green">E49</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>45</sub></font> = <font color="Maroon">c<sub>44</sub></font> as    <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="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>45</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>42</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>43</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>44</sub></font>
;<br/>

<b>thus </b><a NAME="E11:21_3_1_1_9"/>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font>
 ;<br/>

<b>thus </b><a NAME="E12:21_3_1_1_9"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>39</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>42</sub></font>,<font color="Maroon">c<sub>43</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_9"><i><font color="Green">E48</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:21_3_1_1_9"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>44</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>22</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_9"><i><font color="Green">E50</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>46</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E14:21_3_1_1_9"/>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>44</sub></font>
 ;<br/>

<b>hence </b><a NAME="E15:21_3_1_1_9"/>
ex <font color="Olive">b<sub>1</sub></font>, <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#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>46</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>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>46</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>42</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_9"><i><font color="Green">E50</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_9"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>39</sub></font> being    <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> <b> such that </b><br/><a NAME="E52:21_3_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</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>24</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E53:21_3_1_1"/><i><font color="Green">E49</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>24</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>39</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#E50:21_3_1_1"><i><font color="Green">E47</font></i></a>);<br/></i>
<a NAME="E54:21_3_1_1"/><i><font color="Green">E50</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>39</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E55:21_3_1_1"/><i><font color="Green">E51</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>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E56:21_3_1_1"/><b>then </b><i><font color="Green">E52</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>25</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>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E54:21_3_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E57:21_3_1_1"/>
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>25</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>25</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>39</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="21_3_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>40</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_10"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>40</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>25</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_10"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>40</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>25</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="E3:21_3_1_1_10"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>40</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>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E54:21_3_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="polynom6.html#E55:21_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_10"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:21_3_1_1_10"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_10"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E5:21_3_1_1_10"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E55:21_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_10"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:21_3_1_1_10"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>40</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>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green">E56</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>43</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>44</sub></font>, <font color="Maroon">c<sub>45</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E8:21_3_1_1_10"/>
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>42</sub></font> )
 <b>and </b><br/><a NAME="E9:21_3_1_1_10"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font>
 <b>and </b><br/><a NAME="E10:21_3_1_1_10"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font> <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>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E11:21_3_1_1_10"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>43</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1_10"/>
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>43</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#NM1">bag</a> of <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>22</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="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="finseq_4.html#K4">/.</a> <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>44</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>46</sub></font>, <font color="Maroon">c<sub>47</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>48</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="E14:21_3_1_1_10"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font>
 <b>and </b><br/><a NAME="E15:21_3_1_1_10"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</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>46</sub></font>,<font color="Maroon">c<sub>47</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E16:21_3_1_1_10"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>48</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_10"/>
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>48</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>22</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>48</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>46</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>47</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#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green">E56</font></i></a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green">E56</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>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E54:21_3_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_10"><i><font color="Green">E55</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_10"><i><font color="Green">E59</font></i></a>, <a class="txt" href="polynom6.html#E11:21_3_1_1_10"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E14:21_3_1_1_10"><i><font color="Green">E61</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_10"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E19:21_3_1_1_10"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>40</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>39</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_3_1_1_10"><i><font color="Green">E57</font></i></a>, <a class="txt" href="polynom6.html#E6:21_3_1_1_10"><i><font color="Green">E58</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="E58:21_3_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E56:21_3_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E59:21_3_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>29</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#E3:21_3_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E60:21_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</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>28</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E61:21_3_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="polynom6.html#E59:21_3_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="polynom6.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E62:21_3_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</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>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E58:21_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="polynom1.html#T31">POLYNOM1:31</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>40</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E63:21_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<a NAME="E64:21_3_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>40</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>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E62:21_3_1_1"><i><font color="Green">E56</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>41</sub></font>, <font color="Maroon">c<sub>42</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><b> such that </b><br/><a NAME="E66:21_3_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font> <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>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E67:21_3_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>41</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>42</sub></font>
;<br/>

<b>thus </b><a NAME="E68:21_3_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>40</sub></font> <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>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E66:21_3_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>43</sub></font>, <font color="Maroon">c<sub>44</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E70:21_3_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>44</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>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E71:21_3_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>40</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>25</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>43</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>44</sub></font>
 <b>and </b><br/><a NAME="E72:21_3_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</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>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E63:21_3_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>45</sub></font>, <font color="Maroon">c<sub>46</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E74:21_3_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>39</sub></font> &amp; <font color="Maroon">c<sub>46</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>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E75:21_3_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>40</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>39</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>45</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>46</sub></font>
 <b>and </b><br/><a NAME="E76:21_3_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E61:21_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="polynom6.html#E64:21_3_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>44</sub></font> = 
<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>25</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>43</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>44</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>46</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="polynom6.html#E75:21_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>
;<br/>
<a NAME="E78:21_3_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>46</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E58:21_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E74:21_3_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polynom3.html#T22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>47</sub></font>, <font color="Maroon">c<sub>48</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>49</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>50</sub></font>, <font color="Maroon">c<sub>51</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E80:21_3_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>47</sub></font>
 <b>and </b><br/><a NAME="E81:21_3_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>48</sub></font>
 <b>and </b><br/><a NAME="E82:21_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>and </b><br/><a NAME="E83:21_3_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>43</sub></font> <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>47</sub></font>,<font color="Maroon">c<sub>48</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E84:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>49</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E85:21_3_1_1"/><i><font color="Green">E71</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>49</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#NM1">bag</a> of <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>22</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="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <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>50</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>52</sub></font>, <font color="Maroon">c<sub>53</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>54</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="E87:21_3_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>and </b><br/><a NAME="E88:21_3_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>43</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>52</sub></font>,<font color="Maroon">c<sub>53</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E89:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>54</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>22</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E90:21_3_1_1"/><i><font color="Green">E74</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>54</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>22</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>54</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>52</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>53</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#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E91:21_3_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E87:21_3_1_1"><i><font color="Green">E72</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>55</sub></font>, <font color="Maroon">c<sub>56</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="E93:21_3_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>44</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>55</sub></font>,<font color="Maroon">c<sub>56</sub></font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E94:21_3_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>44</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>52</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>55</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E74:21_3_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="polynom6.html#E78:21_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom6.html#E90:21_3_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E95:21_3_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E82:21_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E96:21_3_1_1"/>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>49</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E82:21_3_1_1"><i><font color="Green">E69</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>57</sub></font>, <font color="Maroon">c<sub>58</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E98:21_3_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">c<sub>44</sub></font> <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>57</sub></font>,<font color="Maroon">c<sub>58</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E99:21_3_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>50</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>57</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>58</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E85:21_3_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E64:21_3_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="polynom6.html#E66:21_3_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="polynom6.html#E76:21_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="polynom6.html#E78:21_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>55</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>53</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>56</sub></font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom6.html#E74:21_3_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="polynom6.html#E78:21_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom6.html#E91:21_3_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="polynom6.html#E94:21_3_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E101:21_3_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>53</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>56</sub></font> )
 
<b>by </b><i><a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>59</sub></font> = <font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>60</sub></font> = <font color="Maroon">c<sub>42</sub></font> 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="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>61</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>62</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>, <font color="Maroon">c<sub>63</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E104:21_3_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>61</sub></font>
 <b>and </b><br/><a NAME="E105:21_3_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>59</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>61</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>62</sub></font>
 <b>and </b><br/><a NAME="E106:21_3_1_1"/><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>59</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>63</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>62</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E107:21_3_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>61</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>62</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>55</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E101:21_3_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="polynom6.html#E105:21_3_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E108:21_3_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>57</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>58</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>56</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E93:21_3_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="polynom6.html#E98:21_3_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<a NAME="E109:21_3_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>47</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>52</sub></font> &amp; <font color="Maroon">c<sub>48</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>53</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E83:21_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="polynom6.html#E88:21_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<a NAME="E110:21_3_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>57</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>41</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E80:21_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="polynom6.html#E104:21_3_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="polynom6.html#E106:21_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="polynom6.html#E107:21_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="polynom6.html#E108:21_3_1_1"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>64</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>65</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>, <font color="Maroon">c<sub>66</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E112:21_3_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>64</sub></font>
 <b>and </b><br/><a NAME="E113:21_3_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>60</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>64</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>65</sub></font>
 <b>and </b><br/><a NAME="E114:21_3_1_1"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>60</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>66</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>65</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E115:21_3_1_1"/>
( <font color="Maroon">c<sub>64</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>53</sub></font> &amp; <font color="Maroon">c<sub>65</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>56</sub></font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E101:21_3_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="polynom6.html#E113:21_3_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E116:21_3_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>58</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a>, <a class="txt" href="polynom6.html#E81:21_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom6.html#E108:21_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="polynom6.html#E109:21_3_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="polynom6.html#E112:21_3_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="polynom6.html#E114:21_3_1_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>26</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>40</sub></font> = 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E63:21_3_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>41</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E70:21_3_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="polynom6.html#E72:21_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="polynom6.html#E95:21_3_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="polynom6.html#E99:21_3_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="polynom6.html#E110:21_3_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="polynom6.html#E116:21_3_1_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="group_6.html#V1">multiplicative</a>
 <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>


</div><b>end;</b></div>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</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> holds <span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <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>;<br/>

<b>consider </b><font color="Maroon">c<sub>11</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>12</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>, <font color="Maroon">c<sub>13</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:21_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E3:21_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:21_4"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:21_4"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_4_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</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> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_1"><b>suppose </b></a><a NAME="E1:21_4_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</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/><div class="add"><a NAME="E2:21_4_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:21_4_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self">Th5</a></i>;<br/><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_4_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>hence </b><a NAME="E6:21_4_1_1"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2"><b>suppose </b></a><a NAME="E1:21_4_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</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/><div class="add"><a NAME="E2:21_4_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_4_1_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_4_1_2_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K27">1_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>hence </b><a NAME="E4:21_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_4_1_2_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>hence </b><a NAME="E4:21_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_4_1_2"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E9:21"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
;<br/>
<b>hence </b><a NAME="E11:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="endalg.html#V1">unity-preserving</a>
 <b>by </b><i><a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>;<br/>

<b>thus </b><a NAME="E12:21"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:21_6"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:21_6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:21_6"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</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>11</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E1:21_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E2:21_6"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E8:21_6"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E9:21_6"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polynom6.html#E8:21_6"><i><font color="Green">E25</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</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>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</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>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>23</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>24</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>, <font color="Maroon">c<sub>25</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:21_6_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E3:21_6_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <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>and </b><br/><a NAME="E4:21_6_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_6"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</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>27</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>, <font color="Maroon">c<sub>28</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:21_6_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E7:21_6_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E8:21_6_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_6"><i><font color="Green">E25</font></i></a>, <a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E9:21_6_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_6_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E7:21_6_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E10:21_6_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_6"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E2:21_6_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E4:21_6_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E6:21_6_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E8:21_6_1_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_6_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:21_6"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E13:21"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="polynom6.html#E6:21"><i><font color="Green">E19</font></i></a>, <a class="txt" href="polynom6.html#E11:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="quofield.html#D21">QUOFIELD:def 21</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a><br/>

<b>thus </b><a NAME="E14:21"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E15:21"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</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>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <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>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <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><br/>for <font color="Olive">b<sub>3</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="Olive">b<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font> ) );<br/>
<a NAME="E3:21_7"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</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>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="21_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</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> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>] means for <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E2:21_7_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">S<sub>3</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="21_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/>

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <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>;<br/>

<b>assume </b><a NAME="E1:21_7_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:21_7_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:21_7_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E2:21_7_1"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</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="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>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/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E6:21_7_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</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>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 <font color="Maroon">S<sub>4</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="21_7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <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>;<br/>

<b>consider </b><font color="Maroon">c<sub>17</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>18</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="E2:21_7_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <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> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E4:21_7_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>17</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#E2:21_7_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:21_7_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</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>, <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:21_7_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E6:21_7_1"><i><font color="Green">E25</font></i></a>);<br/></i>
<a NAME="E9:21_7_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:21_7_1_3"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</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="txt" href="polynom6.html#E1:21_7_1_3"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<a NAME="E4:21_7_1_3"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:21_7_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polynom6.html#E2:21_7_1_3"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E6:21_7_1_3"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</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="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="E8:21_7_1_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E9:21_7_1_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_7_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E10:21_7_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:21_7_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom6.html#E9:21_7_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E11:21_7_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E6:21_7_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:21_7_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:21_7_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E14:21_7_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>19</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>20</sub></font> be    <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>;<br/>



<b>assume </b><a NAME="E15:21_7_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>hence </b><a NAME="E16:21_7_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:21_7"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E3:21_7"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM6">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</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="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>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/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>2</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> );<br/>
<a NAME="E7:21_7"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</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>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> st <font color="Maroon">S<sub>3</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="21_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <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>;<br/>

<b>consider </b><font color="Maroon">c<sub>15</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>16</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="E2:21_7_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <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> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E4:21_7_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_7_2"><i><font color="Green">E25</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:21_7_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</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>, <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:21_7"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</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> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E7:21_7"><i><font color="Green">E24</font></i></a>);<br/></i>
<a NAME="E10:21_7"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <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> ;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:21_7_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:21_7_3"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</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><br/>for <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="Olive">b<sub>2</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>17</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="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_7"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E6:21_7_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_7_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E7:21_7_3"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21_7_3"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom6.html#E7:21_7_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom2.html#T19">POLYNOM2:19</a></i>;<br/>
<b>then consider </b><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="E11:21_7_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E12:21_7_3"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <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="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:21_7_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_7_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E14:21_7_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_7_3"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E15:21_7_3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</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>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="E17:21_7_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E18:21_7_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_7"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E19:21_7_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<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>
 
<b>by </b><i><a class="txt" href="polynom6.html#E17:21_7_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polynom6.html#E18:21_7_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E20:21_7_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E12:21_7_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom6.html#E15:21_7_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:21_7"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E12:21_7"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E11:21_7"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E15:21_7"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_7_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <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>;<br/><b>consider </b><font color="Maroon">c<sub>18</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>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>, <font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:21_7_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E3:21_7_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E4:21_7_4"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self">Def2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:21_7_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E7:21_7_4"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</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><br/>for <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="Olive">b<sub>2</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>21</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="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_7"><i><font color="Green">E23</font></i></a></i>;<br/><b>thus </b><a NAME="E8:21_7_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_7_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom6.html#E3:21_7_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="polynom6.html#E4:21_7_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polynom6.html#E6:21_7_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="polynom6.html#E7:21_7_4"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><font color="Maroon">c<sub>10</sub></font> = 
 <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E17</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E18:21_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="polynom6.html#E15:21_7"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>


</div>
