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

<b>let </b><font color="Maroon">o1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">o2</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">L</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">P2</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">P1</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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> ] <b>means </b>for <font color="Olive">P</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">P</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">P</font>;<br/>
<a NAME="E1:21"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span>  ex <font color="Olive">u</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">u</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">x</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">u</font> =  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">Q</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">u</font>
;<br/>

<b>let </b><font color="Maroon">P</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span>;<br/>

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

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


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</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">E37</font></i>: 
<font color="Maroon">f</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">x</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">y</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x'</font>, <font color="Maroon">q</font> = <font color="Maroon">y'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">fp</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">fq</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>, <font color="Maroon">fpq</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">fp</font> = <font color="Maroon">fp</font>, <font color="Maroon">fq</font> = <font color="Maroon">fq</font>, <font color="Maroon">fpq</font> = <font color="Maroon">fpq</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">x</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> holds  <font color="Maroon">fpq</font> <a href="polynom1.html#K15">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="polynom1.html#K15">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Olive">x</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">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>;<br/>

<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:21_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E4:21_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:21_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Q1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E8:21_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E9:21_2_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1''</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:21_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Q1''</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font>
 <b>and </b><br/><a NAME="E12:21_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">b'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1''</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2''</font>
 <b>and </b><br/><a NAME="E13:21_2_1"/><i><font color="Green">E59</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">p</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2''</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> ;<br/>
<a NAME="E15:21_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1''</font> &amp; <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1''</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2'</font> &amp; <font color="Maroon">b2'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2''</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2''</font> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:21_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</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">Q1''</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">Q1'</font>
 
<b>by </b><i>, , , , <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">E64</font></i>: 
<font color="Maroon">Q1''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</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">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><font color="Maroon">fpq</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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">p</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font>
<b>by </b><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">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">fp</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">fq</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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">E65</font></i>: 
<font color="Maroon">f</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">x</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">y</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x'</font>, <font color="Maroon">q</font> = <font color="Maroon">y'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">fp</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">fq</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">fp</font> = <font color="Maroon">fp</font>, <font color="Maroon">fq</font> = <font color="Maroon">fq</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</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">fpq'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">E66</font></i>: 
for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>  ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">fpq'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</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</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Olive">b</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</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</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>;<br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">c</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E3:21_3_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> ;<br/>
<b>consider </b><font color="Maroon">r</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">o2</font>,<font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E6:21_3_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E7:21_3_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</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">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:21_3_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> holds <br/> ex <font color="Olive">a1'</font>, <font color="Olive">b1'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1'</font>,<font color="Olive">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1'</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> ] <b>means </b> ex <font color="Olive">a1'</font>, <font color="Olive">b1'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> ex <font color="Olive">Fk</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ex <font color="Olive">pa1'</font>, <font color="Olive">qb1'</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">pa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1'</font> &amp; <font color="Olive">qb1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1'</font> &amp; <font color="Olive">Fk</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">b1</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">a1'</font>,<font color="Olive">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Fk</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">b2</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> ) ) );<br/>
<a NAME="E9:21_3_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">r</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="finseq_2.html#K3">*</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1"/>
<font color="Maroon">k</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">r</font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1"/><b>then </b>
<font color="Maroon">k</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">b1</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">a1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E4:21_3_1_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">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pa1''</font> = <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1'</font>, <font color="Maroon">qb1''</font> = <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</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">o2</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">pa1'</font> = <font color="Maroon">pa1''</font>, <font color="Maroon">qb1'</font> = <font color="Maroon">qb1''</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</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> ] <b>means </b> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</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">a1''</font>,<font color="Olive">b1''</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">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> );<br/>
<a NAME="E7:21_3_1_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1_1"/>
<font color="Maroon">k</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">b2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1_1"/><b>then </b>
<font color="Maroon">k</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">b2</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">a1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_1_1"/><i><font color="Green">E75</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1''</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1''</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">a1''</font>
;<br/>

<b>take </b>
<font color="Maroon">b1''</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">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:21_3_1_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</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">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:21_3_1_1_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">Fk</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">x</font>
;<br/>

<b>take </b>
<font color="Maroon">a1'</font>
;<br/>

<b>take </b>
<font color="Maroon">b1'</font>
;<br/>

<b>take </b>
<font color="Maroon">Fk</font>
;<br/>

<b>take </b>
<font color="Maroon">pa1'</font>
;<br/>

<b>take </b>
<font color="Maroon">qb1'</font>
;<br/>

<b>thus </b><a NAME="E12:21_3_1_1_1"/>
( <font color="Maroon">pa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">qb1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E14:21_3_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</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">b2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">m</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">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 ;<br/>

<b>hence </b><a NAME="E16:21_3_1_1_1"/>
 ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E11:21_3_1_1"/><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</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">r</font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<b>take </b><font color="Maroon">s</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font>;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">c1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">c2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:21_3_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E16:21_3_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c2</font>
 <b>and </b><br/><a NAME="E17:21_3_1_1"/><i><font color="Green">E86</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">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c2</font>
 <b>by </b><i/>;<br/>
<a NAME="E18:21_3_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">Sr</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

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

<a NAME="E2:21_3_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</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">r</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">o2</font>,<font color="Maroon">L</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">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <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">rFF</font> = <font color="Maroon">r</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">rFF</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">b2</font>;<br/>
<b>consider </b><font color="Maroon">rFFb2</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">E89</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">rFFb2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E24:21_3_1_1"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</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">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E26:21_3_1_1"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</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">rFFb2</font> = <font color="Maroon">rFFb2</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, , <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">rFFb2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_3_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">rFFb2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</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">E134</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rFFb2</font>
 <b>and </b><br/><a NAME="E4:21_3_1_1_3"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">E137</font></i>: 
<font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E1:21"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E6:21_3_1_1_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">rFF</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E1:21"><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">rFF</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">o2</font>,<font color="Maroon">L</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">E141</font></i>: 
<font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <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">E142</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>
 
<b>by </b><i>, <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">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</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">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E5:21"><i><font color="Green">E37</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">rFFb2</font> = <font color="Maroon">rFFb2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">E143</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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>;<br/>
<i><font color="Green">E144</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rFFb2</font> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
<b>by </b><i>, , <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">F</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:21"><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">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rFFb2</font> holds <br/><font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_5"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">rFFb2</font>
 ;<br/>

<b>consider </b><font color="Maroon">c1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">d1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E3:21_3_1_1_5"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">d1</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">E150</font></i>: 
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">d1</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E5:21"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:21_3_1_1_5"/>
<font color="Maroon">k</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">r</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E5:21"><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">a1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">pa1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font>, <font color="Maroon">qb1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:21_3_1_1_5"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">pa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">qb1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> )
 <b>and </b><br/><a NAME="E8:21_3_1_1_5"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E9:21_3_1_1_5"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</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">E181</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E11:21_3_1_1_5"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">ki</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">ki</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">ki</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">ki</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E12:21_3_1_1_5"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">c1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">d1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> )
 
<b>by </b><i>, , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<i><font color="Green">E184</font></i>: <font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="polynom6.html#E5:21"><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">pa1'</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">qb1'</font>
<b>by </b><i>, , <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<b>consider </b><font color="Maroon">s1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:21_3_1_1_5"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">qb1'</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E16:21_3_1_1_5"/><i><font color="Green">E186</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_5"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">ki</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">ki</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font> holds <br/> ex <font color="Olive">x1</font>, <font color="Olive">y2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">ki</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">y2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">ki</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">x1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">y2</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">ki</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_3_1_1_5_2"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">ki</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">s1</font></span>)</span>
 ;<br/><a NAME="E2:21_3_1_1_5_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<font color="Maroon">ki</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</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">x1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font>, <font color="Maroon">y2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_5_2"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y2</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">E191</font></i>: 
<font color="Maroon">s1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">x1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:21_3_1_1_5_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">ki</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 <b>by </b><i>, , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">a1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E8:21_3_1_1_5_2"/><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</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">E194</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:21_3_1_1_5_2"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">x1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1''</font> &amp; <font color="Maroon">y2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1''</font> )
 <b>by </b><i>, , <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">s1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ki</font>
 <b>by </b><i>, <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">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ki</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ki</font>
 <b>by </b><i>, , , , <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">E196</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>thus </b><font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">rFF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, , <a class="txt" href="polynom6.html#E5:21"><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">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E5:21"><i><font color="Green">E37</font></i></a>, , , , <a class="ref" href="polynom6.html#D1" target="_self">Def1</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">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E5:21"><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">E197</font></i>: 
<font color="Maroon">rFFb2</font> <a href="hidden.html#R1">=</a>  <a href="matrlin.html#K8">Sum</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><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">gg</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">o2</font>,<font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E35:21_3_1_1"/><i><font color="Green">E198</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</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">r</font></span>)</span>
 <b>and </b><br/><a NAME="E36:21_3_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">gg</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">o2</font>,<font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E37:21_3_1_1"/><i><font color="Green">E200</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</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">o2</font>,<font color="Maroon">L</font></span>)</span>  st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">j</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">gg</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</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> ] <b>means </b>for <font color="Olive">pp</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</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">r</font> &amp; <font color="Olive">pp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</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">pp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>;<br/>
<a NAME="E38:21_3_1_1"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</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">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">pp'</font> = <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">pp'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>;<br/>

<b>let </b><font color="Maroon">pp</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:21_3_1_1_6"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> &amp; <font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 ;<br/>

<b>hence </b><a NAME="E3:21_3_1_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">ff</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">L</font><b> such that </b><br/><a NAME="E40:21_3_1_1"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E41:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">rFFb2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <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">E203</font></i>: 
<font color="Maroon">Sr</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</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">rFFb2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E43:21_3_1_1"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">gg</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">o2</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <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">r</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">E205</font></i>: <font color="Maroon">ff</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">o2</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</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">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rFFb2</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">j</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">ff</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:21_3_1_1_8"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">rFFb2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rFFb2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</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">E207</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">r</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">pp</font> = <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font>, <font color="Maroon">pp'</font> = <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font>, <font color="Maroon">pp1</font> = <font color="Maroon">pp</font>, <font color="Maroon">pp1'</font> = <font color="Maroon">pp'</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">o2</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w1</font>, <font color="Maroon">pp1</font> = <font color="Maroon">pp1</font>, <font color="Maroon">pp1'</font> = <font color="Maroon">pp1'</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">o2</font>,<font color="Maroon">L</font></span>)</span> ;<br/>
<a NAME="E6:21_3_1_1_8"/><i><font color="Green">E208</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</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">E209</font></i>: 
<font color="Maroon">j</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">r</font>
 
<b>by </b><i>, <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">E210</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <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">j</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">r</font>
 
<b>by </b><i>, , <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">E212</font></i>: 
<font color="Maroon">w</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:21_3_1_1_8"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">j</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">r</font>
 
<b>by </b><i>, <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">E214</font></i>: 
<font color="Maroon">pp1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font>
 
<b>by </b><i>, , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">pp'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">w</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</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">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E47:21_3_1_1"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">Sr</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">rFFb2</font>
 
<b>by </b><i>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E49:21_3_1_1"/>
<font color="Maroon">fpq'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="txt" href="polynom6.html#E5:21"><i><font color="Green">E37</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> ] <b>means </b> ex <font color="Olive">a1'</font>, <font color="Olive">b1'</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">Fk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive">Fk</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">b1</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">a1'</font>,<font color="Olive">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Fk</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">b2</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Olive">Fk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</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">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> ) ) );<br/>
<a NAME="E50:21_3_1_1"/><i><font color="Green">E216</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">r</font></span>)</span> holds <br/> ex <font color="Olive">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">i</font>,<font color="Olive">x</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9"/>
<font color="Maroon">k</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">r</font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9"/><b>then </b>
<font color="Maroon">k</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">b1</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">a1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E4:21_3_1_1_9"/><i><font color="Green">E217</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1'</font> = <font color="Maroon">a1'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b1'</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</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> ] <b>means </b> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</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">a1''</font>,<font color="Olive">b1''</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">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</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">E218</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9_1"/>
<font color="Maroon">k</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">b2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9_1"/><b>then </b>
<font color="Maroon">k</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">b2</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">a1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_9_1"/><i><font color="Green">E219</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1''</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1''</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1''</font> = <font color="Maroon">a1''</font>, <font color="Maroon">b1''</font> = <font color="Maroon">b1''</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">a1''</font>
;<br/>

<b>take </b>
<font color="Maroon">b1''</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">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">x</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">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Fk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:21_3_1_1_9"/><i><font color="Green">E220</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</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">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:21_3_1_1_9"/><i><font color="Green">E221</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>,<font color="Maroon">Fk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">Fk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">x</font>
;<br/>

<b>take </b>
<font color="Maroon">a1'</font>
;<br/>

<b>take </b>
<font color="Maroon">b1'</font>
;<br/>

<b>take </b>
<font color="Maroon">Fk</font>
;<br/>

<b>thus </b><a NAME="E11:21_3_1_1_9"/>
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E13:21_3_1_1_9"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</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">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">m</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">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 ;<br/>

<b>hence </b><a NAME="E15:21_3_1_1_9"/>
 ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Fk</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">m</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">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">E222</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</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">r</font></span>)</span>
 <b>and </b><br/><a NAME="E53:21_3_1_1"/><i><font color="Green">E223</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<a NAME="E54:21_3_1_1"/><i><font color="Green">E224</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</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">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E55:21_3_1_1"/><i><font color="Green">E225</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">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</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"/><i><font color="Green">E227</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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E57:21_3_1_1"/><b>then </b><i><font color="Green">E228</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">F</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">G</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E58:21_3_1_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</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">F</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">j</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_10"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">j</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">F</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_10"/><b>then </b><i><font color="Green">E230</font></i>: 
<font color="Maroon">j</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">F</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">E231</font></i>: 
<font color="Maroon">j</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">G</font></span>)</span>
 
<b>by </b><i>, , , , , <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">E232</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <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">E233</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , , , <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">E234</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">pa1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font>, <font color="Maroon">qb1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:21_3_1_1_10"/>
( <font color="Maroon">pa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1'</font> &amp; <font color="Maroon">qb1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> )
 <b>and </b><br/><a NAME="E9:21_3_1_1_10"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</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">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</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">E236</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1_10"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">a2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Gk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:21_3_1_1_10"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">Gk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</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">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a2'</font>,<font color="Maroon">b2'</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">E238</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Gk</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_10"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gk</font> holds <br/> ex <font color="Olive">a2''</font>, <font color="Olive">b2''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a2''</font>,<font color="Olive">b2''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Gk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</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">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a2''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i>, , </i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <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">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , , , , <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">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:21_3_1_1"/><b>then </b><i><font color="Green">E240</font></i>: 
 <a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E60:21_3_1_1"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">c1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">c2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E61:21_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</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">c1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E62:21_3_1_1"/><b>then </b><i><font color="Green">E242</font></i>: 
 <a href="polynom1.html#K21">decomp</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, , </i>;<br/>
<b>hence </b><a NAME="E63:21_3_1_1"/><i><font color="Green">E243</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</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</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T31">POLYNOM1:31</a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E64:21_3_1_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E65:21_3_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">k</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</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>, <font color="Maroon">c2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E67:21_3_1_1"/><i><font color="Green">E246</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E68:21_3_1_1"/>
<font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/>
<b>take </b>
<font color="Maroon">c1</font>
;<br/>

<b>take </b>
<font color="Maroon">c2</font>
;<br/>

<b>thus </b><a NAME="E69:21_3_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i/>;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E71:21_3_1_1"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">j</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">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>and </b><br/><a NAME="E72:21_3_1_1"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E73:21_3_1_1"/><i><font color="Green">E249</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">i'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E75:21_3_1_1"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j'</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">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> )
 <b>and </b><br/><a NAME="E76:21_3_1_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j'</font>
 <b>and </b><br/><a NAME="E77:21_3_1_1"/><i><font color="Green">E252</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</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#K7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j'</font>
<b>by </b><i>, <a class="txt" href="polynom6.html#E6:21"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>
;<br/>
<a NAME="E79:21_3_1_1"/><b>then </b><i><font color="Green">E253</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font> )
 
<b>by </b><i>, , , <a class="ref" href="polynom3.html#T22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">pa1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font>, <font color="Maroon">qb1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E81:21_3_1_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">pa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E82:21_3_1_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">qb1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E83:21_3_1_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E84:21_3_1_1"/><i><font color="Green">E257</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E85:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E86:21_3_1_1"/><i><font color="Green">E259</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1''</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">ga1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">gb1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Gk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E88:21_3_1_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">Gk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E89:21_3_1_1"/><i><font color="Green">E261</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">ga1'</font>,<font color="Maroon">gb1'</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E90:21_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Gk</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">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E91:21_3_1_1"/><i><font color="Green">E265</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gk</font> holds <br/> ex <font color="Olive">ga1''</font>, <font color="Olive">gb1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">ga1''</font>,<font color="Olive">gb1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Gk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</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">ga1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gb1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">gb1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E92:21_3_1_1"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">Gk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">ga1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">gb1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E94:21_3_1_1"/><i><font color="Green">E267</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">ga1''</font>,<font color="Maroon">gb1''</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E95:21_3_1_1"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">Gk</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</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">ga1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gb1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">gb1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green">E67</font></i></a>, </i>;<br/>
<a NAME="E96:21_3_1_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E97:21_3_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green">E70</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">a1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E99:21_3_1_1"/><i><font color="Green">E270</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E100:21_3_1_1"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1''</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_3_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="txt" href="polynom6.html#E7:21_3_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green">E67</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">ga1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gb1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">gb1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
<b>by </b><i>, <a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green">E67</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E102:21_3_1_1"/><b>then </b><i><font color="Green">E272</font></i>: 
( <font color="Maroon">c1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">ga1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">ga1''</font> &amp; <font color="Maroon">c2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">gb1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">gb1''</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">cc1</font> = <font color="Maroon">c1</font>, <font color="Maroon">cc2</font> = <font color="Maroon">c2</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">cb1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">cb2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E105:21_3_1_1"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">cb1</font>
 <b>and </b><br/><a NAME="E106:21_3_1_1"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">cc1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">cb1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">cb2</font>
 <b>and </b><br/><a NAME="E107:21_3_1_1"/><i><font color="Green">E275</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">cc1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">cb2</font>
 <b>by </b><i/>;<br/>
<a NAME="E108:21_3_1_1"/><i><font color="Green">E276</font></i>: 
( <font color="Maroon">cb1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">ga1'</font> &amp; <font color="Maroon">cb2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">ga1''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green">E84</font></i></a>, </i>;<br/>
<a NAME="E109:21_3_1_1"/><i><font color="Green">E277</font></i>: 
( <font color="Maroon">a1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">ga1''</font> &amp; <font color="Maroon">b1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">gb1''</font> )
 
<b>by </b><i>, , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<a NAME="E110:21_3_1_1"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">a1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">ga1'</font> &amp; <font color="Maroon">b1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">gb1'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1"><i><font color="Green">E71</font></i></a>, , <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<a NAME="E111:21_3_1_1"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">pa1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">a1''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E6:21_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom6.html#E15:21_3_1_1"><i><font color="Green">E83</font></i></a>, , <a class="txt" href="polynom6.html#E17:21_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="polynom6.html#E18:21_3_1_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">cb1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">cb2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E113:21_3_1_1"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">cb1</font>
 <b>and </b><br/><a NAME="E114:21_3_1_1"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">cc2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">cb1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">cb2</font>
 <b>and </b><br/><a NAME="E115:21_3_1_1"/><i><font color="Green">E282</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">cc2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">cb2</font>
 <b>by </b><i/>;<br/>
<a NAME="E116:21_3_1_1"/>
( <font color="Maroon">cb1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">gb1'</font> &amp; <font color="Maroon">cb2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">gb1''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="polynom6.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E117:21_3_1_1"/><b>then </b><i><font color="Green">E283</font></i>: 
<font color="Maroon">qb1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c2</font>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E18:21_3_1_1"><i><font color="Green">E87</font></i></a>, , <a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green">E90</font></i></a>, </i>;<br/>
<b>thus </b><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">fq</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c2</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green">E89</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
<font color="Maroon">fp</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">fq</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</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">f</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">1P1</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">1P2</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">E334</font></i>: 
for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</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">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:21_4"/><i><font color="Green">E335</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E3:21_4"/><i><font color="Green">E336</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E4:21_4"/><i><font color="Green">E337</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:21_4"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> or <font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">E339</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_4_1_1"/><b>then </b><i><font color="Green">E340</font></i>: 
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:21_4_1_1"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font>
 <b>by </b><i>, , </i>;<br/><font color="Maroon">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</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> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
;<br/><b>then </b><font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</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">L</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, , <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">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><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">E342</font></i>: 
<font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_4_1_2"/><b>then </b><i><font color="Green">E343</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font> or <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o2</font> )
 <b>by </b><i>, </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">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font> or <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</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">E344</font></i>: 
<font color="Maroon">b1</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><font color="Maroon">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</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> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K27">1_</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><b>then </b><font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, , <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, , <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">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><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">E345</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><font color="Maroon">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <font color="Maroon">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
<b>by </b><i>, <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">o2</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>then </b><font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
<br/>.= 
<font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, , <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">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><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">1P1</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">1P2</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<font color="Maroon">f</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">o1</font>,<span class="p3">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">1P1</font>
<b>by </b><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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <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">E346</font></i>: 
<font color="Maroon">f</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">f</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">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</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">E348</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:21_6"/><i><font color="Green">E349</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>assume </b><a NAME="E3:21_6"/><i><font color="Green">E350</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x1</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1''</font> = <font color="Maroon">x1'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">x2'</font> = <font color="Maroon">x2</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2''</font> = <font color="Maroon">x2'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">E355</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">x2''</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:21_6"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1'</font> <a href="hidden.html#R1">=</a>  <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">x1''</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">w2</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><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">b1</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>;<br/><b>reconsider </b><font color="Maroon">x1''b1</font> = <font color="Maroon">x1''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>, <font color="Maroon">x2''b1</font> = <font color="Maroon">x2''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</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">b2</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>;<br/><b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:21_6_1_1"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font>
 <b>and </b><br/><a NAME="E3:21_6_1_1"/><i><font color="Green">E360</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E4:21_6_1_1"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">w1</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">c1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">c2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1'</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:21_6_1_1"/><i><font color="Green">E362</font></i>: 
<font color="Maroon">Q1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c1</font>
 <b>and </b><br/><a NAME="E7:21_6_1_1"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c2</font>
 <b>and </b><br/><a NAME="E8:21_6_1_1"/><i><font color="Green">E364</font></i>: 
<font color="Maroon">w2</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:21_6_1_1"/>
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2'</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">b1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">b2'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font> )
 <b>by </b><i>, <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E10:21_6_1_1"/>
<font color="Maroon">x1''b1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2''b1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_6_1"/>
<font color="Maroon">x1''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</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">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</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">f</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i>, , , <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">f</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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">y</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">y</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">y</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</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">o1</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">o2</font>,<font color="Maroon">L</font></span>)</span>] <b>means </b> ex <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><br/> for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  st <font color="Olive">b</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">b2</font> holds <br/><font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font> ) );<br/>
<a NAME="E3:21_7"/><i><font color="Green">E367</font></i>: 
for <font color="Olive">x</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>  ex <font color="Olive">y</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">o2</font>,<font color="Maroon">L</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</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">x</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>;<br/>

<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</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">o2</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b>for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  st <font color="Olive">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</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">s</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font>;<br/>
<a NAME="E2:21_7_1"/><i><font color="Green">E368</font></i>: 
for <font color="Olive">p</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>  ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</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">p</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/>

<b>take </b><font color="Maroon">q</font> = <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">p</font></span>)</span>;<br/>

<b>let </b><font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:21_7_1_1"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><a NAME="E2:21_7_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">t</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:21_7_1"/><i><font color="Green">E369</font></i>: 
for <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b2</font>,<font color="Maroon">t</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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> ] <b>means </b> ex <font color="Olive">b1</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font> );<br/>
<a NAME="E6:21_7_1"/><i><font color="Green">E371</font></i>: 
for <font color="Olive">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  ex <font color="Olive">y</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</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">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E2:21_7_1_2"/><i><font color="Green">E372</font></i>: 
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b2</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">b1</font>
;<br/>

<b>take </b>
<font color="Maroon">b2</font>
;<br/>

<b>thus </b><a NAME="E4:21_7_1_2"/>
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:21_7_1_2"/>
<font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">kk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>, <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E8:21_7_1"/><i><font color="Green">E374</font></i>: 
for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b</font>,<font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E9:21_7_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">kk</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</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">x</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">E375</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:21_7_1_3"/><b>then </b><i><font color="Green">E376</font></i>: 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>;<br/>
<a NAME="E4:21_7_1_3"/><i><font color="Green">E377</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">s</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:21_7_1_3"/><b>then </b><i><font color="Green">E378</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E8:21_7_1_3"/><i><font color="Green">E379</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E9:21_7_1_3"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:21_7_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, , </i>;<br/>
<b>hence </b><a NAME="E11:21_7_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">kk</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , <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">t</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">t</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</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">t''</font> = <font color="Maroon">t</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">o2</font>,<font color="Maroon">L</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">t'</font> = <font color="Maroon">t</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon">t''</font>
;<br/>

<b>take </b>
<font color="Maroon">t'</font>
;<br/>

<b>thus </b><a NAME="E14:21_7_1"/>
<font color="Maroon">t''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t'</font>
 ;<br/>

<b>let </b><font color="Maroon">b2</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>



<b>assume </b><a NAME="E15:21_7_1"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">x</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 ;<br/>

<b>hence </b><a NAME="E16:21_7_1"/>
<font color="Maroon">t'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</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">o2</font>,<font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E5:21_7"/><i><font color="Green">E384</font></i>: 
for <font color="Olive">x</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM5">Function</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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> ] <b>means </b> ex <font color="Olive">b1</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> );<br/>
<a NAME="E7:21_7"/><i><font color="Green">E385</font></i>: 
for <font color="Olive">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  ex <font color="Olive">y</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</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">x</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E2:21_7_2"/><i><font color="Green">E386</font></i>: 
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b1</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">b1</font>
;<br/>

<b>take </b>
<font color="Maroon">b2</font>
;<br/>

<b>thus </b><a NAME="E4:21_7_2"/>
<font color="Maroon">x</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:21_7_2"/>
<font color="Maroon">y</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">kk</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>, <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font><b> such that </b><br/><a NAME="E9:21_7"/><i><font color="Green">E387</font></i>: 
for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b</font>,<font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E10:21_7"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">kk</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</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">x</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">E388</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">x</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ;<br/>
<b>consider </b><font color="Maroon">h</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">E389</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E5:21_7_3"/><i><font color="Green">E390</font></i>: 
for <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><br/> for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  st <font color="Olive">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:21_7_3"/>
<font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</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">o2</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <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">E391</font></i>: 
<font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font> <b>by </b><i>, <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">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="polynom2.html#T19">POLYNOM2:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E11:21_7_3"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">h</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">b</font> = <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>;<br/>
<a NAME="E12:21_7_3"/><i><font color="Green">E393</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">kk</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</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">h</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <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">s</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:21_7_3"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E17:21_7_3"/><i><font color="Green">E395</font></i>: 
<font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2'</font>
 <b>and </b><br/><a NAME="E18:21_7_3"/><i><font color="Green">E396</font></i>: 
<font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font>
 <b>by </b><i/>;<br/>
<a NAME="E19:21_7_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E20:21_7_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">kk</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</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">E397</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">g</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">g</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">g</font> = <font color="Maroon">g</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">g'</font> = <font color="Maroon">g</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i>, <a class="ref" href="polynom1.html#D10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E15:21_7"/><i><font color="Green">E399</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">o2</font>,<font color="Maroon">L</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">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>;<br/><b>consider </b><font color="Maroon">b1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">Q1</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">o2</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:21_7_4"/><i><font color="Green">E400</font></i>: 
<font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E3:21_7_4"/><i><font color="Green">E401</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E4:21_7_4"/><i><font color="Green">E402</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">g'</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">h</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">E403</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E7:21_7_4"/><i><font color="Green">E404</font></i>: 
for <font color="Olive">b2</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><br/> for <font color="Olive">b</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">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>  st <font color="Olive">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:21_7_4"/>
<font color="Maroon">s</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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">g'</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<b>then </b><font color="Maroon">y</font> = 
 <a href="polynom6.html#K2" target="_self">Compress</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E18:21_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


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


</div>
