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

<b>let </b><font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c2">o2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>



<b>set </b><font color="Maroon" title="c4">P2</font> =  <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">P1</font> =  <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">P</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Olive" title="b1">P</font>;<br/>
<a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c6">x</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">u</font> =  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c7">Q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">u</font>
;<br/>

<b>let </b><font color="Maroon" title="c9">P</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:21_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P</font>
 ;<br/>

<b>hence </b><a NAME="E4:21_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c9">P</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span><b> such that </b><br/><a NAME="E3:21"/><i><font color="Green" title="E17">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D26" title="QUOFIELD:def.26">QUOFIELD:def 26</a><br/>

<b>thus </b><a NAME="E5:21"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a>,<a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a>,<a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">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" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a><br/>

<b>reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y'</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c10">x'</font>, <font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c11">y'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">fp</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c15">fq</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c16">fpq</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">fp</font> = <font color="Maroon" title="c14">fp</font>, <font color="Maroon" title="c18">fq</font> = <font color="Maroon" title="c15">fq</font>, <font color="Maroon" title="c19">fpq</font> = <font color="Maroon" title="c16">fpq</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E5:21_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> holds  <font color="Maroon" title="c19">fpq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">fp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">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" title="c20">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c21">b'</font> = <font color="Maroon" title="c20">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c23">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c24">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:21_2_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c24">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b1</font>
 <b>and </b><br/><a NAME="E4:21_2_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c21">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c23">b2</font>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green" title="E21">A6</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c26">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c27">Q1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E7:21_2_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c27">Q1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c25">b1'</font>
 <b>and </b><br/><a NAME="E8:21_2_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c21">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c26">b2'</font>
 <b>and </b><br/><a NAME="E9:21_2_1"/><i><font color="Green" title="E24">A9</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Q1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c26">b2'</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">b1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c29">b2''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c30">Q1''</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E11:21_2_1"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c30">Q1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c28">b1''</font>
 <b>and </b><br/><a NAME="E12:21_2_1"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c21">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b1''</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c29">b2''</font>
 <b>and </b><br/><a NAME="E13:21_2_1"/><i><font color="Green" title="E27">A12</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Q1''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c29">b2''</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">b1</font> = <font color="Maroon" title="c22">b1</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/>
<a NAME="E15:21_2_1"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c31">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">b1'</font> &amp; <font color="Maroon" title="c25">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b1''</font> &amp; <font color="Maroon" title="c31">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">b1''</font> &amp; <font color="Maroon" title="c23">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">b2'</font> &amp; <font color="Maroon" title="c26">b2'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">b2''</font> &amp; <font color="Maroon" title="c23">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">b2''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_2_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="polynom6.html#E8:21_2_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="polynom6.html#E12:21_2_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E16:21_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c31">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c31">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c31">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>;<br/>
<a NAME="E17:21_2_1"/><b>then </b>
<font color="Maroon" title="c30">Q1''</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c24">Q1</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c27">Q1'</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_2_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="polynom6.html#E7:21_2_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="polynom6.html#E11:21_2_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="polynom6.html#E15:21_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E18:21_2_1"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c30">Q1''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">Q1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>;<br/>
<a NAME="E19:21_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c13">q</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c19">fpq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font> = 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b'</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b'</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polynom6.html#E5:21_2_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="polynom6.html#E9:21_2_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="polynom6.html#E13:21_2_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="polynom6.html#E15:21_2_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="polynom6.html#E18:21_2_1"><i><font color="Green" title="E29">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">fp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y</font></span>)</span> = 
<font color="Maroon" title="c17">fp</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c18">fq</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E6:21"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">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" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y'</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c10">x'</font>, <font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c11">y'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">fp</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c15">fq</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c16">fp</font> = <font color="Maroon" title="c14">fp</font>, <font color="Maroon" title="c17">fq</font> = <font color="Maroon" title="c15">fq</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:21_3_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">fpq'</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E7:21_3_1"/><i><font color="Green" title="E20">A16</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c18">fpq'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b2">s</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">s</font> holds <br/> ex <font color="Olive" title="b4">b1</font>, <font color="Olive" title="b5">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b2">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">fp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b4">b1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b5">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" title="c19">c</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c19">c</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c22">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E3:21_3_1_1"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c20">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c22">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">b1</font> = <font color="Maroon" title="c21">b1</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/>
<b>consider </b><font color="Maroon" title="c24">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><b> such that </b><br/><a NAME="E6:21_3_1_1"/><i><font color="Green" title="E22">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c24">r</font>
 <b>and </b><br/><a NAME="E7:21_3_1_1"/><i><font color="Green" title="E23">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:21_3_1_1"/><i><font color="Green" title="E24">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font> holds <br/> ex <font color="Olive" title="b2">a1'</font>, <font color="Olive" title="b3">b1'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a1'</font>,<font color="Olive" title="b3">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c24">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">a1'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b1'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a1'</font>, <font color="Olive" title="b2">b1'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b4">pa1'</font>, <font color="Olive" title="b5">qb1'</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b4">pa1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">a1'</font> &amp; <font color="Olive" title="b5">qb1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1'</font> &amp; <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a1'</font>,<font color="Olive" title="b2">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Fk</font> holds <br/> ex <font color="Olive" title="b7">a1''</font>, <font color="Olive" title="b8">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b7">a1''</font>,<font color="Olive" title="b8">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b3">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b7">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b8">b1''</font></span>)</span> ) ) );<br/>
<a NAME="E9:21_3_1_1"/><i><font color="Green" title="E25">A21</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">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" title="c25">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1"/>
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">a1'</font>, <font color="Maroon" title="c27">b1'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E4:21_3_1_1_1"/><i><font color="Green" title="E26">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c26">a1'</font>,<font color="Maroon" title="c27">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c23">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">a1'</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c27">b1'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">pa1''</font> = <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c26">a1'</font>, <font color="Maroon" title="c29">qb1''</font> = <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c27">b1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c30">pa1'</font> = <font color="Maroon" title="c28">pa1''</font>, <font color="Maroon" title="c31">qb1'</font> = <font color="Maroon" title="c29">qb1''</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a1''</font>, <font color="Olive" title="b2">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a1''</font>,<font color="Olive" title="b2">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1''</font></span>)</span> );<br/>
<a NAME="E7:21_3_1_1_1"/><i><font color="Green" title="E27">A23</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">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" title="c32">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_1_1"/>
<font color="Maroon" title="c32">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c32">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c33">a1''</font>, <font color="Maroon" title="c34">b1''</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_1_1"/><i><font color="Green" title="E28">A24</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c32">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">a1''</font>,<font color="Maroon" title="c34">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c22">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c33">a1''</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c34">b1''</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c35">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c30">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c33">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c34">b1''</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c35">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c33">a1''</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c34">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c32">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">a1''</font>,<font color="Maroon" title="c34">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c33">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c34">b1''</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_1_1"><i><font color="Green" title="E28">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c32">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E9:21_3_1_1_1"/><i><font color="Green" title="E28">A25</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:21_3_1_1_1"/><i><font color="Green" title="E29">A26</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c32">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E7:21_3_1_1_1"><i><font color="Green" title="E27">A23</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c33">x</font> = <font color="Maroon" title="c32">Fk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c33">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c26">a1'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c27">b1'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c32">Fk</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c30">pa1'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c31">qb1'</font>
;<br/>

<b>thus </b><a NAME="E12:21_3_1_1_1"/>
( <font color="Maroon" title="c30">pa1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c26">a1'</font> &amp; <font color="Maroon" title="c31">qb1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c27">b1'</font> &amp; <font color="Maroon" title="c32">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c26">a1'</font>,<font color="Maroon" title="c27">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_1"><i><font color="Green" title="E26">A22</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:21_3_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_3_1_1_1"><i><font color="Green" title="E28">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon" title="c34">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E15:21_3_1_1_1"/>
<font color="Maroon" title="c34">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">Fk</font>
 ;<br/>

<b>hence </b><a NAME="E16:21_3_1_1_1"/>
 ex <font color="Olive" title="b1">a1''</font>, <font color="Olive" title="b2">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">a1''</font>,<font color="Olive" title="b2">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c32">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1''</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_3_1_1_1"><i><font color="Green" title="E28">A25</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_1"><i><font color="Green" title="E29">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c25">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E11:21_3_1_1"/><i><font color="Green" title="E26">A27</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1"/><i><font color="Green" title="E27">A28</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c25">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E9:21_3_1_1"><i><font color="Green" title="E25">A21</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c26">s</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c25">F</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c27">x</font> = <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">c1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c29">c2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c30">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E15:21_3_1_1"/><i><font color="Green" title="E28">A29</font></i>: 
<font color="Maroon" title="c30">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c28">c1</font>
 <b>and </b><br/><a NAME="E16:21_3_1_1"/><i><font color="Green" title="E29">A30</font></i>: 
<font color="Maroon" title="c20">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c29">c2</font>
 <b>and </b><br/><a NAME="E17:21_3_1_1"/><i><font color="Green" title="E30">A31</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c29">c2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<a NAME="E18:21_3_1_1"/><i><font color="Green" title="E31">A32</font></i>: 
( <font color="Maroon" title="c23">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c28">c1</font> &amp; <font color="Maroon" title="c22">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">c2</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">Sr</font> =  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c24">r</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E20:21_3_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font> holds <br/><font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">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" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_2"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font>
 ;<br/>

<a NAME="E2:21_3_1_1_2"/><b>then </b><i><font color="Green" title="E32">A33</font></i>: 
<font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:21_3_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:21_3_1_1_2"/>
<font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_2"><i><font color="Green" title="E32">A33</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c32">rFF</font> = <font color="Maroon" title="c24">r</font> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">b2</font>;<br/>
<b>consider </b><font color="Maroon" title="c33">rFFb2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E23:21_3_1_1"/><i><font color="Green" title="E32">A34</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c33">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font>
 <b>and </b><br/><a NAME="E24:21_3_1_1"/><i><font color="Green" title="E33">A35</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font> holds <br/><font color="Maroon" title="c33">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c34">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E26:21_3_1_1"/><i><font color="Green" title="E34">A36</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c34">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c35">rFFb2</font> = <font color="Maroon" title="c33">rFFb2</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E26:21_3_1_1"><i><font color="Green" title="E34">A36</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E28:21_3_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c35">rFFb2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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" title="c36">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_3_1_1_3"/>
<font color="Maroon" title="c36">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c35">rFFb2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c37">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_3_1_1_3"/><i><font color="Green" title="E35">A37</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c35">rFFb2</font>
 <b>and </b><br/><a NAME="E4:21_3_1_1_3"/><i><font color="Green" title="E36">A38</font></i>: 
<font color="Maroon" title="c36">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:21_3_1_1_3"/><i><font color="Green" title="E37">A39</font></i>: 
<font color="Maroon" title="c35">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">b2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_3"><i><font color="Green" title="E35">A37</font></i></a></i>;<br/>
<a NAME="E6:21_3_1_1_3"/><i><font color="Green" title="E38">A40</font></i>: 
<font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">rFF</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_3"><i><font color="Green" title="E35">A37</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:21_3_1_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">rFF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:21_3_1_1_3"/><b>then </b><i><font color="Green" title="E39">A41</font></i>: 
<font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_3"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21_3_1_1_3"/><b>then </b><i><font color="Green" title="E40">A42</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:21_3_1_1_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_3"><i><font color="Green" title="E39">A41</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E12:21_3_1_1_3"/>
<font color="Maroon" title="c36">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_3"><i><font color="Green" title="E36">A38</font></i></a>, <a class="txt" href="polynom6.html#E5:21_3_1_1_3"><i><font color="Green" title="E37">A39</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_3"><i><font color="Green" title="E40">A42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c36">rFFb2</font> = <font color="Maroon" title="c35">rFFb2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E30:21_3_1_1"/><i><font color="Green" title="E35">A43</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></i>;<br/>
<i><font color="Green" title="E36">A44</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">rFFb2</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">F</font>
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E30:21_3_1_1"><i><font color="Green" title="E35">A43</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
;<br/>
<a NAME="E32:21_3_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">rFFb2</font> holds <br/><font color="Maroon" title="c36">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c37">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_5"/><i><font color="Green" title="E37">A45</font></i>: 
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">rFFb2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c38">c1</font>, <font color="Maroon" title="c39">d1</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E3:21_3_1_1_5"/><i><font color="Green" title="E38">A46</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c38">c1</font>,<font color="Maroon" title="c39">d1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:21_3_1_1_5"/><i><font color="Green" title="E39">A47</font></i>: 
<font color="Maroon" title="c24">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c38">c1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c39">d1</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a></i>;<br/>
<a NAME="E5:21_3_1_1_5"/>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c40">a1'</font>, <font color="Maroon" title="c41">b1'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c42">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c43">pa1'</font>, <font color="Maroon" title="c44">qb1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E7:21_3_1_1_5"/><i><font color="Green" title="E40">A48</font></i>: 
( <font color="Maroon" title="c43">pa1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c40">a1'</font> &amp; <font color="Maroon" title="c44">qb1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c41">b1'</font> )
 <b>and </b><br/><a NAME="E8:21_3_1_1_5"/><i><font color="Green" title="E41">A49</font></i>: 
<font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c37">k</font>
 <b>and </b><br/><a NAME="E9:21_3_1_1_5"/><i><font color="Green" title="E42">A50</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c40">a1'</font>,<font color="Maroon" title="c41">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E10:21_3_1_1_5"/><i><font color="Green" title="E43">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E11:21_3_1_1_5"/><i><font color="Green" title="E44">A52</font></i>: 
for <font color="Olive" title="b1">ki</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font>, <font color="Olive" title="b3">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a1''</font>,<font color="Olive" title="b3">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c42">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b1''</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green" title="E27">A28</font></i></a></i>;<br/>
<a NAME="E12:21_3_1_1_5"/><i><font color="Green" title="E45">A53</font></i>: 
( <font color="Maroon" title="c38">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c40">a1'</font> &amp; <font color="Maroon" title="c39">d1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c41">b1'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_3_1_1_5"><i><font color="Green" title="E38">A46</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_5"><i><font color="Green" title="E42">A50</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<i><font color="Green" title="E46">A54</font></i>: <font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> = 
<font color="Maroon" title="c24">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c44">qb1'</font>
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_5"><i><font color="Green" title="E39">A47</font></i></a>, <a class="txt" href="polynom6.html#E7:21_3_1_1_5"><i><font color="Green" title="E40">A48</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1_5"><i><font color="Green" title="E45">A53</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c45">s1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E15:21_3_1_1_5"/><i><font color="Green" title="E47">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c44">qb1'</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c45">s1</font>
 <b>and </b><br/><a NAME="E16:21_3_1_1_5"/><i><font color="Green" title="E48">A56</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_5"/><i><font color="Green" title="E49">A57</font></i>: 
for <font color="Olive" title="b1">ki</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">s1</font> holds <br/> ex <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">y2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">y2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c45">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">x1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">y2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">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" title="c46">ki</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:21_3_1_1_5_2"/><i><font color="Green" title="E50">A58</font></i>: 
<font color="Maroon" title="c46">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c45">s1</font></span>)</span>
 ;<br/><a NAME="E2:21_3_1_1_5_2"/><b>then </b><i><font color="Green" title="E51">A59</font></i>: 
<font color="Maroon" title="c46">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c45">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c47">x1</font>, <font color="Maroon" title="c48">y2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_5_2"/><i><font color="Green" title="E52">A60</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c47">x1</font>,<font color="Maroon" title="c48">y2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:21_3_1_1_5_2"/><i><font color="Green" title="E53">A61</font></i>: 
<font color="Maroon" title="c45">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c47">x1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c48">y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E17:21_3_1_1_5"><i><font color="Green" title="E49">A57</font></i></a></i>;<br/><a NAME="E6:21_3_1_1_5_2"/><i><font color="Green" title="E54">A62</font></i>: 
<font color="Maroon" title="c46">ki</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Fk</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:21_3_1_1_5"><i><font color="Green" title="E43">A51</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_5"><i><font color="Green" title="E48">A56</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5_2"><i><font color="Green" title="E50">A58</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c49">a1''</font>, <font color="Maroon" title="c50">b1''</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E8:21_3_1_1_5_2"/><i><font color="Green" title="E55">A63</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c49">a1''</font>,<font color="Maroon" title="c50">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E9:21_3_1_1_5_2"/><i><font color="Green" title="E56">A64</font></i>: 
<font color="Maroon" title="c42">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c49">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c50">b1''</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1_5"><i><font color="Green" title="E44">A52</font></i></a></i>;<br/><a NAME="E10:21_3_1_1_5_2"/><i><font color="Green" title="E57">A65</font></i>: 
( <font color="Maroon" title="c47">x1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c49">a1''</font> &amp; <font color="Maroon" title="c48">y2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c50">b1''</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_5_2"><i><font color="Green" title="E52">A60</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_5_2"><i><font color="Green" title="E55">A63</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/><a NAME="E11:21_3_1_1_5_2"/>
<font color="Maroon" title="c45">s1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">ki</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_5_2"><i><font color="Green" title="E51">A59</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E12:21_3_1_1_5_2"/>
<font color="Maroon" title="c45">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">ki</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">ki</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_3_1_1_5_2"><i><font color="Green" title="E53">A61</font></i></a>, <a class="txt" href="polynom6.html#E6:21_3_1_1_5_2"><i><font color="Green" title="E54">A62</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_5_2"><i><font color="Green" title="E56">A64</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_5_2"><i><font color="Green" title="E57">A65</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E50">A66</font></i>: 
<font color="Maroon" title="c45">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Fk</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:21_3_1_1_5"><i><font color="Green" title="E43">A51</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_5"><i><font color="Green" title="E48">A56</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c36">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">rFF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">b2</font>
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_5"><i><font color="Green" title="E41">A49</font></i></a>, <a class="txt" href="polynom6.html#E13:21_3_1_1_5"><i><font color="Green" title="E46">A54</font></i></a>, <a class="txt" href="polynom6.html#E15:21_3_1_1_5"><i><font color="Green" title="E47">A55</font></i></a>, <a class="txt" href="polynom6.html#E19:21_3_1_1_5"><i><font color="Green" title="E50">A66</font></i></a>, <a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_5"><i><font color="Green" title="E37">A45</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="E37">A67</font></i>: 
<font color="Maroon" title="c36">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E31:21_3_1_1"><i><font color="Green" title="E36">A44</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c37">gg</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><b> such that </b><br/><a NAME="E35:21_3_1_1"/><i><font color="Green" title="E38">A68</font></i>: 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 <b>and </b><br/><a NAME="E36:21_3_1_1"/><i><font color="Green" title="E39">A69</font></i>: 
<font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <b>and </b><br/><a NAME="E37:21_3_1_1"/><i><font color="Green" title="E40">A70</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">v</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">pp</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>  st $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Olive" title="b1">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>;<br/>
<a NAME="E38:21_3_1_1"/><i><font color="Green" title="E41">A71</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c38">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c39">pp'</font> = <font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">x</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>take </b><font color="Maroon" title="c40">y</font> = <font color="Maroon" title="c39">pp'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>;<br/>

<b>let </b><font color="Maroon" title="c41">pp</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>;<br/>

<b>assume </b><a NAME="E2:21_3_1_1_6"/>
( <font color="Maroon" title="c38">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c41">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">x</font> )
 ;<br/>

<b>hence </b><a NAME="E3:21_3_1_1_6"/>
<font color="Maroon" title="c40">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">pp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c38">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E40:21_3_1_1"/><i><font color="Green" title="E42">A72</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E38:21_3_1_1"><i><font color="Green" title="E41">A71</font></i></a>);<br/></i>
<a NAME="E41:21_3_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">rFFb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E42:21_3_1_1"/><b>then </b><i><font color="Green" title="E43">A73</font></i>: 
<font color="Maroon" title="c31">Sr</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">rFFb2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E35:21_3_1_1"><i><font color="Green" title="E38">A68</font></i></a>, <a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green" title="E42">A72</font></i></a></i>;<br/>
<a NAME="E43:21_3_1_1"/><i><font color="Green" title="E44">A74</font></i>: 
<font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E36:21_3_1_1"><i><font color="Green" title="E39">A69</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E44:21_3_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E45">A75</font></i>: <font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green" title="E42">A72</font></i></a>, <a class="txt" href="polynom6.html#E43:21_3_1_1"><i><font color="Green" title="E44">A74</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/>
<a NAME="E46:21_3_1_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">rFFb2</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">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" title="c39">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c40">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>;<br/>



<b>assume </b><a NAME="E1:21_3_1_1_8"/><i><font color="Green" title="E46">A76</font></i>: 
( <font color="Maroon" title="c39">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">rFFb2</font> &amp; <font color="Maroon" title="c40">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">rFFb2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:21_3_1_1_8"/><b>then </b><i><font color="Green" title="E47">A77</font></i>: 
<font color="Maroon" title="c39">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E23:21_3_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c41">w</font> = <font color="Maroon" title="c24">r</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c42">pp</font> = <font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">j</font>, <font color="Maroon" title="c43">pp'</font> = <font color="Maroon" title="c37">gg</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c44">w1</font> = <font color="Maroon" title="c41">w</font>, <font color="Maroon" title="c45">pp1</font> = <font color="Maroon" title="c42">pp</font>, <font color="Maroon" title="c46">pp1'</font> = <font color="Maroon" title="c43">pp'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c47">w1</font> = <font color="Maroon" title="c44">w1</font>, <font color="Maroon" title="c48">pp1</font> = <font color="Maroon" title="c45">pp1</font>, <font color="Maroon" title="c49">pp1'</font> = <font color="Maroon" title="c46">pp1'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<a NAME="E6:21_3_1_1_8"/><i><font color="Green" title="E48">A78</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E7:21_3_1_1_8"/><i><font color="Green" title="E49">A79</font></i>: 
<font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green" title="E47">A77</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:21_3_1_1_8"/><b>then </b><i><font color="Green" title="E50">A80</font></i>: 
<font color="Maroon" title="c41">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_8"><i><font color="Green" title="E48">A78</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:21_3_1_1_8"/>
<font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1_8"><i><font color="Green" title="E48">A78</font></i></a>, <a class="txt" href="polynom6.html#E7:21_3_1_1_8"><i><font color="Green" title="E49">A79</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:21_3_1_1_8"/><b>then </b><i><font color="Green" title="E51">A81</font></i>: 
<font color="Maroon" title="c41">w</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">v</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E24:21_3_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_8"><i><font color="Green" title="E46">A76</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_8"><i><font color="Green" title="E50">A80</font></i></a></i>;<br/>
<a NAME="E11:21_3_1_1_8"/><i><font color="Green" title="E52">A82</font></i>: 
<font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green" title="E47">A77</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E12:21_3_1_1_8"/><i><font color="Green" title="E53">A83</font></i>: 
<font color="Maroon" title="c49">pp1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">pp1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c47">w1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E37:21_3_1_1"><i><font color="Green" title="E40">A70</font></i></a>, <a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green" title="E47">A77</font></i></a>, <a class="txt" href="polynom6.html#E8:21_3_1_1_8"><i><font color="Green" title="E50">A80</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c43">pp'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green" title="E42">A72</font></i></a>, <a class="txt" href="polynom6.html#E11:21_3_1_1_8"><i><font color="Green" title="E52">A82</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c42">pp</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c41">w</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font>
<b>by </b><i><a class="txt" href="polynom6.html#E12:21_3_1_1_8"><i><font color="Green" title="E53">A83</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c42">pp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">w</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">ff</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">j</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c40">v</font>
<b>by </b><i><a class="txt" href="polynom6.html#E40:21_3_1_1"><i><font color="Green" title="E42">A72</font></i></a>, <a class="txt" href="polynom6.html#E2:21_3_1_1_8"><i><font color="Green" title="E47">A77</font></i></a>, <a class="txt" href="polynom6.html#E10:21_3_1_1_8"><i><font color="Green" title="E51">A81</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E47:21_3_1_1"/><b>then </b><i><font color="Green" title="E46">A84</font></i>: 
<font color="Maroon" title="c31">Sr</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c36">rFFb2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E42:21_3_1_1"><i><font color="Green" title="E43">A73</font></i></a>, <a class="txt" href="polynom6.html#E45:21_3_1_1"><i><font color="Green" title="E45">A75</font></i></a>, <a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>;<br/>
<a NAME="E48:21_3_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E49:21_3_1_1"/>
<font color="Maroon" title="c18">fpq'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c26">s</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:21_3_1_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="txt" href="polynom6.html#E15:21_3_1_1"><i><font color="Green" title="E28">A29</font></i></a>, <a class="txt" href="polynom6.html#E17:21_3_1_1"><i><font color="Green" title="E30">A31</font></i></a>, <a class="txt" href="polynom6.html#E18:21_3_1_1"><i><font color="Green" title="E31">A32</font></i></a>, <a class="txt" href="polynom6.html#E33:21_3_1_1"><i><font color="Green" title="E37">A67</font></i></a>, <a class="txt" href="polynom6.html#E47:21_3_1_1"><i><font color="Green" title="E46">A84</font></i></a>, <a class="ref" href="polynom1.html#T34" title="POLYNOM1:th.34">POLYNOM1:34</a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a1'</font>, <font color="Olive" title="b2">b1'</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b3">Fk</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a1'</font>,<font color="Olive" title="b2">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> &amp; ( for <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">Fk</font> holds <br/> ex <font color="Olive" title="b5">a1''</font>, <font color="Olive" title="b6">b1''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b5">a1''</font>,<font color="Olive" title="b6">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b3">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b5">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b6">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> ) ) );<br/>
<a NAME="E50:21_3_1_1"/><i><font color="Green" title="E47">A85</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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" title="c39">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9"/>
<font color="Maroon" title="c39">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9"/><b>then </b>
<font color="Maroon" title="c39">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c40">a1'</font>, <font color="Maroon" title="c41">b1'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E4:21_3_1_1_9"/><i><font color="Green" title="E48">A86</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c39">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c40">a1'</font>,<font color="Maroon" title="c41">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c23">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c40">a1'</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c41">b1'</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c42">a1'</font> = <font color="Maroon" title="c40">a1'</font>, <font color="Maroon" title="c43">b1'</font> = <font color="Maroon" title="c41">b1'</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">a1''</font>, <font color="Olive" title="b2">b1''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a1''</font>,<font color="Olive" title="b2">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c42">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> );<br/>
<a NAME="E6:21_3_1_1_9"/><i><font color="Green" title="E49">A87</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">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" title="c44">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_9_1"/>
<font color="Maroon" title="c44">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_9_1"/><b>then </b>
<font color="Maroon" title="c44">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c45">a1''</font>, <font color="Maroon" title="c46">b1''</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E4:21_3_1_1_9_1"/><i><font color="Green" title="E50">A88</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c44">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c45">a1''</font>,<font color="Maroon" title="c46">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c22">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c45">a1''</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c46">b1''</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c47">a1''</font> = <font color="Maroon" title="c45">a1''</font>, <font color="Maroon" title="c48">b1''</font> = <font color="Maroon" title="c46">b1''</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c49">x</font> = <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c42">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c47">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c48">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c49">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c47">a1''</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c48">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c44">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c47">a1''</font>,<font color="Maroon" title="c48">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c49">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c42">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c47">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c48">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_9_1"><i><font color="Green" title="E50">A88</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c44">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:21_3_1_1_9"/><i><font color="Green" title="E50">A89</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:21_3_1_1_9"/><i><font color="Green" title="E51">A90</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c44">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E6:21_3_1_1_9"><i><font color="Green" title="E49">A87</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c45">x</font> = <font color="Maroon" title="c44">Fk</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c45">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c42">a1'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c43">b1'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c44">Fk</font>
;<br/>

<b>thus </b><a NAME="E11:21_3_1_1_9"/>
<font color="Maroon" title="c44">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c39">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c42">a1'</font>,<font color="Maroon" title="c43">b1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_9"><i><font color="Green" title="E48">A86</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:21_3_1_1_9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c44">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_9"><i><font color="Green" title="E50">A89</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon" title="c46">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E14:21_3_1_1_9"/>
<font color="Maroon" title="c46">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c44">Fk</font>
 ;<br/>

<b>hence </b><a NAME="E15:21_3_1_1_9"/>
 ex <font color="Olive" title="b1">a1''</font>, <font color="Olive" title="b2">b1''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c46">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a1''</font>,<font color="Olive" title="b2">b1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c44">Fk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c46">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c42">a1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c43">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_3_1_1_9"><i><font color="Green" title="E50">A89</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_9"><i><font color="Green" title="E51">A90</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c39">G</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E52:21_3_1_1"/><i><font color="Green" title="E48">A91</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 <b>and </b><br/><a NAME="E53:21_3_1_1"/><i><font color="Green" title="E49">A92</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c39">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom6.html#E50:21_3_1_1"><i><font color="Green" title="E47">A85</font></i></a>);<br/></i>
<a NAME="E54:21_3_1_1"/><i><font color="Green" title="E50">A93</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c39">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">i</font>]
 
<b>by </b><i><a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green" title="E49">A92</font></i></a></i>;<br/>
<a NAME="E55:21_3_1_1"/><i><font color="Green" title="E51">A94</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">G</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E56:21_3_1_1"/><i><font color="Green" title="E52">A95</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E57:21_3_1_1"/><b>then </b><i><font color="Green" title="E53">A96</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="polynom6.html#E55:21_3_1_1"><i><font color="Green" title="E51">A94</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E58:21_3_1_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">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" title="c40">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_3_1_1_10"/><i><font color="Green" title="E54">A97</font></i>: 
<font color="Maroon" title="c40">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_3_1_1_10"/><b>then </b><i><font color="Green" title="E55">A98</font></i>: 
<font color="Maroon" title="c40">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:21_3_1_1_10"/><i><font color="Green" title="E56">A99</font></i>: 
<font color="Maroon" title="c40">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="polynom6.html#E55:21_3_1_1"><i><font color="Green" title="E51">A94</font></i></a>, <a class="txt" href="polynom6.html#E56:21_3_1_1"><i><font color="Green" title="E52">A95</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_10"><i><font color="Green" title="E54">A97</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:21_3_1_1_10"/><i><font color="Green" title="E57">A100</font></i>: 
<font color="Maroon" title="c40">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_3_1_1_10"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E5:21_3_1_1_10"/><i><font color="Green" title="E58">A101</font></i>: 
<font color="Maroon" title="c40">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">G</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="polynom6.html#E56:21_3_1_1"><i><font color="Green" title="E52">A95</font></i></a>, <a class="txt" href="polynom6.html#E1:21_3_1_1_10"><i><font color="Green" title="E54">A97</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:21_3_1_1_10"/><i><font color="Green" title="E59">A102</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c40">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green" title="E57">A100</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c41">a1'</font>, <font color="Maroon" title="c42">b1'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c43">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c44">pa1'</font>, <font color="Maroon" title="c45">qb1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E8:21_3_1_1_10"/>
( <font color="Maroon" title="c44">pa1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c41">a1'</font> &amp; <font color="Maroon" title="c45">qb1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c42">b1'</font> )
 <b>and </b><br/><a NAME="E9:21_3_1_1_10"/><i><font color="Green" title="E60">A103</font></i>: 
<font color="Maroon" title="c43">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c41">a1'</font>,<font color="Maroon" title="c42">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E11:21_3_1_1_10"/><i><font color="Green" title="E61">A104</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c43">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:21_3_1_1_10"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c43">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font>, <font color="Olive" title="b3">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a1''</font>,<font color="Olive" title="b3">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c43">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b1''</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green" title="E57">A100</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c46">a2'</font>, <font color="Maroon" title="c47">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c48">Gk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:21_3_1_1_10"/><i><font color="Green" title="E62">A105</font></i>: 
<font color="Maroon" title="c48">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">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" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c46">a2'</font>,<font color="Maroon" title="c47">b2'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E16:21_3_1_1_10"/><i><font color="Green" title="E63">A106</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c48">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:21_3_1_1_10"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c48">Gk</font> holds <br/> ex <font color="Olive" title="b2">a2''</font>, <font color="Olive" title="b3">b2''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">a2''</font>,<font color="Olive" title="b3">b2''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c48">Gk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">a2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">a2''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c47">b2'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b2''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green" title="E49">A92</font></i></a>, <a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green" title="E57">A100</font></i></a></i>;<br/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">j</font></span>)</span> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_3_1_1_10"><i><font color="Green" title="E57">A100</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E55:21_3_1_1"><i><font color="Green" title="E51">A94</font></i></a>, <a class="txt" href="polynom6.html#E3:21_3_1_1_10"><i><font color="Green" title="E56">A99</font></i></a>, <a class="txt" href="polynom6.html#E9:21_3_1_1_10"><i><font color="Green" title="E60">A103</font></i></a>, <a class="txt" href="polynom6.html#E11:21_3_1_1_10"><i><font color="Green" title="E61">A104</font></i></a>, <a class="txt" href="polynom6.html#E14:21_3_1_1_10"><i><font color="Green" title="E62">A105</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1_10"><i><font color="Green" title="E63">A106</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c40">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c40">j</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_3_1_1_10"><i><font color="Green" title="E58">A101</font></i></a>, <a class="txt" href="polynom6.html#E6:21_3_1_1_10"><i><font color="Green" title="E59">A102</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">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" title="E54">A107</font></i>: 
 <a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E57:21_3_1_1"><i><font color="Green" title="E53">A96</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E60:21_3_1_1"/><i><font color="Green" title="E55">A108</font></i>: 
( <font color="Maroon" title="c28">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b1</font> &amp; <font color="Maroon" title="c29">c2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b2</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_3_1_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green" title="E29">A30</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E61:21_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c28">c1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E62:21_3_1_1"/><b>then </b><i><font color="Green" title="E56">A109</font></i>: 
 <a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c39">G</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E16:21_3_1_1"><i><font color="Green" title="E29">A30</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="polynom6.html#E54:21_3_1_1"><i><font color="Green" title="E50">A93</font></i></a>, <a class="txt" href="polynom6.html#E60:21_3_1_1"><i><font color="Green" title="E55">A108</font></i></a>, <a class="ref" href="polynom6.html#T14" target="_self" title="POLYNOM6:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E63:21_3_1_1"/><i><font color="Green" title="E57">A110</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E59:21_3_1_1"><i><font color="Green" title="E54">A107</font></i></a>, <a class="ref" href="polynom1.html#T31" title="POLYNOM1:th.31">POLYNOM1:31</a></i>;<br/>

<b>let </b><font color="Maroon" title="c40">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E64:21_3_1_1"/><i><font color="Green" title="E58">A111</font></i>: 
<font color="Maroon" title="c40">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">s</font>
 ;<br/>

<a NAME="E65:21_3_1_1"/><b>then </b><i><font color="Green" title="E59">A112</font></i>: 
<font color="Maroon" title="c40">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E63:21_3_1_1"><i><font color="Green" title="E57">A110</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">c1</font>, <font color="Maroon" title="c42">c2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E67:21_3_1_1"/><i><font color="Green" title="E60">A113</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c41">c1</font>,<font color="Maroon" title="c42">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E68:21_3_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c41">c1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c42">c2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T72" title="POLYNOM1:th.72">POLYNOM1:72</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c41">c1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c42">c2</font>
;<br/>

<b>thus </b><a NAME="E69:21_3_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c40">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c41">c1</font>,<font color="Maroon" title="c42">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E67:21_3_1_1"><i><font color="Green" title="E60">A113</font></i></a></i>;<br/>

<b>consider </b><font color="Maroon" title="c43">i</font>, <font color="Maroon" title="c44">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E71:21_3_1_1"/><i><font color="Green" title="E61">A114</font></i>: 
( <font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">F</font> &amp; <font color="Maroon" title="c44">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span>)</span> )
 <b>and </b><br/><a NAME="E72:21_3_1_1"/><i><font color="Green" title="E62">A115</font></i>: 
<font color="Maroon" title="c40">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c43">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">j</font>
 <b>and </b><br/><a NAME="E73:21_3_1_1"/><i><font color="Green" title="E63">A116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">k</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E64:21_3_1_1"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c45">i'</font>, <font color="Maroon" title="c46">j'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E75:21_3_1_1"/><i><font color="Green" title="E64">A117</font></i>: 
( <font color="Maroon" title="c45">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c39">G</font> &amp; <font color="Maroon" title="c46">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c39">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">i'</font></span>)</span> )
 <b>and </b><br/><a NAME="E76:21_3_1_1"/><i><font color="Green" title="E65">A118</font></i>: 
<font color="Maroon" title="c40">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c39">G</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c45">i'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c46">j'</font>
 <b>and </b><br/><a NAME="E77:21_3_1_1"/><i><font color="Green" title="E66">A119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c45">i'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c46">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c19">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">k</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E62:21_3_1_1"><i><font color="Green" title="E56">A109</font></i></a>, <a class="txt" href="polynom6.html#E65:21_3_1_1"><i><font color="Green" title="E59">A112</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c25">F</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c43">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">j</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c43">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c44">j</font>
<b>by </b><i><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c39">G</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c45">i'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c46">j'</font>
<b>by </b><i><a class="txt" href="polynom6.html#E72:21_3_1_1"><i><font color="Green" title="E62">A115</font></i></a>, <a class="txt" href="polynom6.html#E76:21_3_1_1"><i><font color="Green" title="E65">A118</font></i></a>, <a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></i>
;<br/>
<a NAME="E79:21_3_1_1"/><b>then </b><i><font color="Green" title="E67">A120</font></i>: 
( <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">i'</font> &amp; <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">j'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E59:21_3_1_1"><i><font color="Green" title="E54">A107</font></i></a>, <a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="txt" href="polynom6.html#E75:21_3_1_1"><i><font color="Green" title="E64">A117</font></i></a>, <a class="ref" href="polynom3.html#T22" title="POLYNOM3:th.22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c47">a1'</font>, <font color="Maroon" title="c48">b1'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c49">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c50">pa1'</font>, <font color="Maroon" title="c51">qb1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E81:21_3_1_1"/><i><font color="Green" title="E68">A121</font></i>: 
<font color="Maroon" title="c50">pa1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c47">a1'</font>
 <b>and </b><br/><a NAME="E82:21_3_1_1"/><i><font color="Green" title="E69">A122</font></i>: 
<font color="Maroon" title="c51">qb1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c48">b1'</font>
 <b>and </b><br/><a NAME="E83:21_3_1_1"/><i><font color="Green" title="E70">A123</font></i>: 
<font color="Maroon" title="c49">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">F</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c43">i</font>
 <b>and </b><br/><a NAME="E84:21_3_1_1"/><i><font color="Green" title="E71">A124</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c47">a1'</font>,<font color="Maroon" title="c48">b1'</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E85:21_3_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c49">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E86:21_3_1_1"/><i><font color="Green" title="E72">A125</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">Fk</font> holds <br/> ex <font color="Olive" title="b2">a1''</font>, <font color="Olive" title="b3">b1''</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">a1''</font>,<font color="Olive" title="b3">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c49">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b1''</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E12:21_3_1_1"><i><font color="Green" title="E27">A28</font></i></a>, <a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c52">ga1'</font>, <font color="Maroon" title="c53">gb1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c54">Gk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E88:21_3_1_1"/><i><font color="Green" title="E73">A126</font></i>: 
<font color="Maroon" title="c54">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">G</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c43">i</font>
 <b>and </b><br/><a NAME="E89:21_3_1_1"/><i><font color="Green" title="E74">A127</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c23">b1</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c43">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c52">ga1'</font>,<font color="Maroon" title="c53">gb1'</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E90:21_3_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c54">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E91:21_3_1_1"/><i><font color="Green" title="E75">A128</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c54">Gk</font> holds <br/> ex <font color="Olive" title="b2">ga1''</font>, <font color="Olive" title="b3">gb1''</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">ga1''</font>,<font color="Olive" title="b3">gb1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c54">Gk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c52">ga1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c53">gb1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">gb1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E53:21_3_1_1"><i><font color="Green" title="E49">A92</font></i></a>, <a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a></i>;<br/>
<a NAME="E92:21_3_1_1"/><i><font color="Green" title="E76">A129</font></i>: 
<font color="Maroon" title="c54">Gk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_3_1_1"><i><font color="Green" title="E26">A27</font></i></a>, <a class="txt" href="polynom6.html#E52:21_3_1_1"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="txt" href="polynom6.html#E88:21_3_1_1"><i><font color="Green" title="E73">A126</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c55">ga1''</font>, <font color="Maroon" title="c56">gb1''</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E94:21_3_1_1"/><i><font color="Green" title="E77">A130</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c55">ga1''</font>,<font color="Maroon" title="c56">gb1''</font></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E95:21_3_1_1"/><i><font color="Green" title="E78">A131</font></i>: 
<font color="Maroon" title="c54">Gk</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c52">ga1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c55">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c53">gb1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c56">gb1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom6.html#E75:21_3_1_1"><i><font color="Green" title="E64">A117</font></i></a>, <a class="txt" href="polynom6.html#E79:21_3_1_1"><i><font color="Green" title="E67">A120</font></i></a>, <a class="txt" href="polynom6.html#E91:21_3_1_1"><i><font color="Green" title="E75">A128</font></i></a></i>;<br/>
<a NAME="E96:21_3_1_1"/><i><font color="Green" title="E79">A132</font></i>: 
<font color="Maroon" title="c49">Fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="txt" href="polynom6.html#E83:21_3_1_1"><i><font color="Green" title="E70">A123</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E97:21_3_1_1"/>
<font color="Maroon" title="c44">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c49">Fk</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="txt" href="polynom6.html#E83:21_3_1_1"><i><font color="Green" title="E70">A123</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c57">a1''</font>, <font color="Maroon" title="c58">b1''</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E99:21_3_1_1"/><i><font color="Green" title="E80">A133</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c22">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c57">a1''</font>,<font color="Maroon" title="c58">b1''</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E100:21_3_1_1"/><i><font color="Green" title="E81">A134</font></i>: 
<font color="Maroon" title="c49">Fk</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c44">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c50">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c57">a1''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c58">b1''</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E86:21_3_1_1"><i><font color="Green" title="E72">A125</font></i></a></i>;<br/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c41">c1</font>,<font color="Maroon" title="c42">c2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c39">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">j</font>
<b>by </b><i><a class="txt" href="polynom6.html#E65:21_3_1_1"><i><font color="Green" title="E59">A112</font></i></a>, <a class="txt" href="polynom6.html#E67:21_3_1_1"><i><font color="Green" title="E60">A113</font></i></a>, <a class="txt" href="polynom6.html#E77:21_3_1_1"><i><font color="Green" title="E66">A119</font></i></a>, <a class="txt" href="polynom6.html#E79:21_3_1_1"><i><font color="Green" title="E67">A120</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c52">ga1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c55">ga1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c53">gb1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c56">gb1''</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom6.html#E75:21_3_1_1"><i><font color="Green" title="E64">A117</font></i></a>, <a class="txt" href="polynom6.html#E79:21_3_1_1"><i><font color="Green" title="E67">A120</font></i></a>, <a class="txt" href="polynom6.html#E92:21_3_1_1"><i><font color="Green" title="E76">A129</font></i></a>, <a class="txt" href="polynom6.html#E95:21_3_1_1"><i><font color="Green" title="E78">A131</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E102:21_3_1_1"/><b>then </b><i><font color="Green" title="E82">A135</font></i>: 
( <font color="Maroon" title="c41">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c52">ga1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c55">ga1''</font> &amp; <font color="Maroon" title="c42">c2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c53">gb1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c56">gb1''</font> )
 
<b>by </b><i><a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c59">cc1</font> = <font color="Maroon" title="c41">c1</font>, <font color="Maroon" title="c60">cc2</font> = <font color="Maroon" title="c42">c2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c61">cb1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c62">cb2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c63">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E105:21_3_1_1"/><i><font color="Green" title="E83">A136</font></i>: 
<font color="Maroon" title="c63">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c61">cb1</font>
 <b>and </b><br/><a NAME="E106:21_3_1_1"/><i><font color="Green" title="E84">A137</font></i>: 
<font color="Maroon" title="c59">cc1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c61">cb1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c62">cb2</font>
 <b>and </b><br/><a NAME="E107:21_3_1_1"/><i><font color="Green" title="E85">A138</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c59">cc1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c62">cb2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<a NAME="E108:21_3_1_1"/><i><font color="Green" title="E86">A139</font></i>: 
( <font color="Maroon" title="c61">cb1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c52">ga1'</font> &amp; <font color="Maroon" title="c62">cb2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c55">ga1''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E102:21_3_1_1"><i><font color="Green" title="E82">A135</font></i></a>, <a class="txt" href="polynom6.html#E106:21_3_1_1"><i><font color="Green" title="E84">A137</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E109:21_3_1_1"/><i><font color="Green" title="E87">A140</font></i>: 
( <font color="Maroon" title="c57">a1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c55">ga1''</font> &amp; <font color="Maroon" title="c58">b1''</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c56">gb1''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E94:21_3_1_1"><i><font color="Green" title="E77">A130</font></i></a>, <a class="txt" href="polynom6.html#E99:21_3_1_1"><i><font color="Green" title="E80">A133</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<a NAME="E110:21_3_1_1"/><i><font color="Green" title="E88">A141</font></i>: 
( <font color="Maroon" title="c47">a1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c52">ga1'</font> &amp; <font color="Maroon" title="c48">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c53">gb1'</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E84:21_3_1_1"><i><font color="Green" title="E71">A124</font></i></a>, <a class="txt" href="polynom6.html#E89:21_3_1_1"><i><font color="Green" title="E74">A127</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<a NAME="E111:21_3_1_1"/><b>then </b><i><font color="Green" title="E89">A142</font></i>: 
<font color="Maroon" title="c50">pa1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c57">a1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c41">c1</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polynom6.html#E81:21_3_1_1"><i><font color="Green" title="E68">A121</font></i></a>, <a class="txt" href="polynom6.html#E105:21_3_1_1"><i><font color="Green" title="E83">A136</font></i></a>, <a class="txt" href="polynom6.html#E107:21_3_1_1"><i><font color="Green" title="E85">A138</font></i></a>, <a class="txt" href="polynom6.html#E108:21_3_1_1"><i><font color="Green" title="E86">A139</font></i></a>, <a class="txt" href="polynom6.html#E109:21_3_1_1"><i><font color="Green" title="E87">A140</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c64">cb1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c65">cb2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c66">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E113:21_3_1_1"/><i><font color="Green" title="E90">A143</font></i>: 
<font color="Maroon" title="c66">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c64">cb1</font>
 <b>and </b><br/><a NAME="E114:21_3_1_1"/><i><font color="Green" title="E91">A144</font></i>: 
<font color="Maroon" title="c60">cc2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c64">cb1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c65">cb2</font>
 <b>and </b><br/><a NAME="E115:21_3_1_1"/><i><font color="Green" title="E92">A145</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c60">cc2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c65">cb2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<a NAME="E116:21_3_1_1"/>
( <font color="Maroon" title="c64">cb1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c53">gb1'</font> &amp; <font color="Maroon" title="c65">cb2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c56">gb1''</font> )
 
<b>by </b><i><a class="txt" href="polynom6.html#E102:21_3_1_1"><i><font color="Green" title="E82">A135</font></i></a>, <a class="txt" href="polynom6.html#E114:21_3_1_1"><i><font color="Green" title="E91">A144</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<a NAME="E117:21_3_1_1"/><b>then </b><i><font color="Green" title="E93">A146</font></i>: 
<font color="Maroon" title="c51">qb1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c58">b1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c42">c2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="polynom6.html#E82:21_3_1_1"><i><font color="Green" title="E69">A122</font></i></a>, <a class="txt" href="polynom6.html#E109:21_3_1_1"><i><font color="Green" title="E87">A140</font></i></a>, <a class="txt" href="polynom6.html#E110:21_3_1_1"><i><font color="Green" title="E88">A141</font></i></a>, <a class="txt" href="polynom6.html#E113:21_3_1_1"><i><font color="Green" title="E90">A143</font></i></a>, <a class="txt" href="polynom6.html#E115:21_3_1_1"><i><font color="Green" title="E92">A145</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c26">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c40">k</font> = 
<font color="Maroon" title="c26">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">k</font>
<b>by </b><i><a class="txt" href="polynom6.html#E64:21_3_1_1"><i><font color="Green" title="E58">A111</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">fp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c41">c1</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">fq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c42">c2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E71:21_3_1_1"><i><font color="Green" title="E61">A114</font></i></a>, <a class="txt" href="polynom6.html#E73:21_3_1_1"><i><font color="Green" title="E63">A116</font></i></a>, <a class="txt" href="polynom6.html#E96:21_3_1_1"><i><font color="Green" title="E79">A132</font></i></a>, <a class="txt" href="polynom6.html#E100:21_3_1_1"><i><font color="Green" title="E81">A134</font></i></a>, <a class="txt" href="polynom6.html#E111:21_3_1_1"><i><font color="Green" title="E89">A142</font></i></a>, <a class="txt" href="polynom6.html#E117:21_3_1_1"><i><font color="Green" title="E93">A146</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">y</font></span>)</span> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
<br/>.= 
<font color="Maroon" title="c16">fp</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c17">fq</font>
<b>by </b><i><a class="txt" href="polynom6.html#E7:21_3_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3"/>
<font color="Maroon" title="c7">f</font> is <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>


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

<b>reconsider </b><font color="Maroon" title="c8">1P1</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">1P2</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21"/><i><font color="Green" title="E20">A147</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">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" title="c10">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c11">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c12">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c13">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E2:21_4"/><i><font color="Green" title="E21">A148</font></i>: 
<font color="Maroon" title="c13">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">1P1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c11">b1</font>
 <b>and </b><br/><a NAME="E3:21_4"/><i><font color="Green" title="E22">A149</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c12">b2</font>
 <b>and </b><br/><a NAME="E4:21_4"/><i><font color="Green" title="E23">A150</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/>
<a NAME="E5:21_4"/><i><font color="Green" title="E24">A151</font></i>: 
<font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">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" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> or <font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">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" title="E25">A152</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_4_1_1"/><b>then </b><i><font color="Green" title="E26">A153</font></i>: 
<font color="Maroon" title="c12">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">o2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green" title="E22">A149</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></i>;<br/><a NAME="E3:21_4_1_1"/><i><font color="Green" title="E27">A154</font></i>: 
<font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green" title="E22">A149</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_1"><i><font color="Green" title="E25">A152</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></i>;<br/><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c11">b1</font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green" title="E21">A148</font></i></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21_4_1_1"><i><font color="Green" title="E27">A154</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>
;<br/><b>then </b><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">b2</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">b2</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4_1_1"><i><font color="Green" title="E26">A153</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green" title="E24">A151</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_1"><i><font color="Green" title="E25">A152</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">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" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green" title="E23">A150</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2"><b>suppose </b></a><a NAME="E1:21_4_1_2"/><i><font color="Green" title="E25">A155</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:21_4_1_2"/><b>then </b><i><font color="Green" title="E26">A156</font></i>: 
( <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o1</font> or <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c2">o2</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_4"><i><font color="Green" title="E22">A149</font></i></a>, <a class="ref" href="polynom6.html#T5" target="_self" title="POLYNOM6:th.5">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_4_1_2_1_1"/>
( <font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o1</font> or <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">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" title="E27">A157</font></i>: 
<font color="Maroon" title="c11">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o1</font>
 ;<br/><div class="add"><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c11">b1</font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green" title="E21">A148</font></i></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_1"><i><font color="Green" title="E27">A157</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
;<br/><b>then </b><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">b2</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4_1_2"><i><font color="Green" title="E26">A156</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_1"><i><font color="Green" title="E27">A157</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green" title="E24">A151</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2"><i><font color="Green" title="E25">A155</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">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" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green" title="E23">A150</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_4_1_2_1_1_2"/><i><font color="Green" title="E27">A158</font></i>: 
<font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o1</font>
 ;<br/><div class="add"><font color="Maroon" title="c13">Q1</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c11">b1</font>
<b>by </b><i><a class="txt" href="polynom6.html#E2:21_4"><i><font color="Green" title="E21">A148</font></i></a>, <a class="ref" href="polynom1.html#T90" title="POLYNOM1:th.90">POLYNOM1:90</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_4_1_2_1_1_2"><i><font color="Green" title="E27">A158</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">POLYNOM1:84</a></i>
<br/>.= 
 <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><b>then </b><font color="Maroon" title="c13">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">b2</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
<br/>.= 
<font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_4"><i><font color="Green" title="E24">A151</font></i></a>, <a class="txt" href="polynom6.html#E1:21_4_1_2"><i><font color="Green" title="E25">A155</font></i></a>, <a class="ref" href="polynom1.html#T84" title="POLYNOM1:th.84">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" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_4"><i><font color="Green" title="E23">A150</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:21_4_1_2"/>
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">1P2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c10">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p3">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c8">1P1</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom6.html#E9:21"><i><font color="Green" title="E20">A147</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
;<br/>
<b>hence </b><a NAME="E11:21"/><i><font color="Green" title="E21">A159</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">unity-preserving</a>
 <b>by </b><i><a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a></i>;<br/>

<b>thus </b><a NAME="E12:21"/>
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:21_6"/><i><font color="Green" title="E22">A160</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:21_6"/><i><font color="Green" title="E23">A161</font></i>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>assume </b><a NAME="E3:21_6"/><i><font color="Green" title="E24">A162</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">x1'</font> = <font color="Maroon" title="c10">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E1:21_6"><i><font color="Green" title="E22">A160</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x1''</font> = <font color="Maroon" title="c12">x1'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x2'</font> = <font color="Maroon" title="c11">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E2:21_6"><i><font color="Green" title="E23">A161</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x2''</font> = <font color="Maroon" title="c14">x2'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E8:21_6"/><i><font color="Green" title="E25">A163</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c15">x2''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<a NAME="E9:21_6"/><i><font color="Green" title="E26">A164</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c13">x1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">w1</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">w2</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="polynom6.html#E8:21_6"><i><font color="Green" title="E25">A163</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">b1</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>;<br/><b>reconsider </b><font color="Maroon" title="c19">x1''b1</font> = <font color="Maroon" title="c13">x1''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font>, <font color="Maroon" title="c20">x2''b1</font> = <font color="Maroon" title="c15">x2''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">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" title="c21">b2</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>;<br/><b>set </b><font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font>;<br/><b>consider </b><font color="Maroon" title="c23">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c24">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c25">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E2:21_6_1_1"/><i><font color="Green" title="E27">A165</font></i>: 
<font color="Maroon" title="c25">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c23">b1'</font>
 <b>and </b><br/><a NAME="E3:21_6_1_1"/><i><font color="Green" title="E28">A166</font></i>: 
<font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c24">b2'</font>
 <b>and </b><br/><a NAME="E4:21_6_1_1"/><i><font color="Green" title="E29">A167</font></i>: 
<font color="Maroon" title="c16">w1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c24">b2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_6"><i><font color="Green" title="E26">A164</font></i></a>, <a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">c1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c27">c2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c28">Q1'</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E6:21_6_1_1"/><i><font color="Green" title="E30">A168</font></i>: 
<font color="Maroon" title="c28">Q1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c26">c1</font>
 <b>and </b><br/><a NAME="E7:21_6_1_1"/><i><font color="Green" title="E31">A169</font></i>: 
<font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c27">c2</font>
 <b>and </b><br/><a NAME="E8:21_6_1_1"/><i><font color="Green" title="E32">A170</font></i>: 
<font color="Maroon" title="c17">w2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Q1'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c27">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_6"><i><font color="Green" title="E25">A163</font></i></a>, <a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/><a NAME="E9:21_6_1_1"/>
( <font color="Maroon" title="c18">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">b1'</font> &amp; <font color="Maroon" title="c18">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">c1</font> &amp; <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">b2'</font> &amp; <font color="Maroon" title="c21">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">c2</font> &amp; <font color="Maroon" title="c23">b1'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">c1</font> &amp; <font color="Maroon" title="c24">b2'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c27">c2</font> )
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_6_1_1"><i><font color="Green" title="E28">A166</font></i></a>, <a class="txt" href="polynom6.html#E7:21_6_1_1"><i><font color="Green" title="E31">A169</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E10:21_6_1_1"/>
<font color="Maroon" title="c19">x1''b1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2''b1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c21">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E3:21_6"><i><font color="Green" title="E24">A162</font></i></a>, <a class="txt" href="polynom6.html#E2:21_6_1_1"><i><font color="Green" title="E27">A165</font></i></a>, <a class="txt" href="polynom6.html#E4:21_6_1_1"><i><font color="Green" title="E29">A167</font></i></a>, <a class="txt" href="polynom6.html#E6:21_6_1_1"><i><font color="Green" title="E30">A168</font></i></a>, <a class="txt" href="polynom6.html#E8:21_6_1_1"><i><font color="Green" title="E32">A170</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:21_6_1"/>
<font color="Maroon" title="c13">x1''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2''</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:21_6"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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

<b>thus </b><a NAME="E13:21"/>
<font color="Maroon" title="c7">f</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polynom6.html#E6:21"><i><font color="Green" title="E19">A15</font></i></a>, <a class="txt" href="polynom6.html#E11:21"><i><font color="Green" title="E21">A159</font></i></a>, <a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a><br/>

<b>thus </b><a NAME="E14:21"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E15:21"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c10">y</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>] <b>means </b> ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; ( for <font color="Olive" title="b2">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b3">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b3">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> $1 <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> holds <br/><font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b</font> ) );<br/>
<a NAME="E3:21_7"/><i><font color="Green" title="E22">A171</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c12">x</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">b1</font> = <font color="Maroon" title="c12">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font>] <b>means </b>for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b</font>;<br/>
<a NAME="E2:21_7_1"/><i><font color="Green" title="E23">A172</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>  ex <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">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" title="c14">p</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>;<br/>

<b>take </b><font color="Maroon" title="c15">q</font> = <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">p</font></span>)</span>;<br/>

<b>let </b><font color="Maroon" title="c16">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:21_7_1_1"/>
<font color="Maroon" title="c16">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c14">p</font>
 ;<br/>

<b>hence </b><a NAME="E2:21_7_1_1"/>
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">t</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E4:21_7_1"/><i><font color="Green" title="E24">A173</font></i>: 
for <font color="Olive" title="b1">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b2</font>,<font color="Maroon" title="c14">t</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b2</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E2:21_7_1"><i><font color="Green" title="E23">A172</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c14">t</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b2</font> );<br/>
<a NAME="E6:21_7_1"/><i><font color="Green" title="E25">A174</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c16">x</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c17">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c18">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E2:21_7_1_2"/><i><font color="Green" title="E26">A175</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">b2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c19">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c17">b1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c18">b2</font>
;<br/>

<b>thus </b><a NAME="E4:21_7_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_7_1_2"><i><font color="Green" title="E26">A175</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:21_7_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b2</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c16">kk</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>, <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E8:21_7_1"/><i><font color="Green" title="E26">A176</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c16">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E6:21_7_1"><i><font color="Green" title="E25">A174</font></i></a>);<br/></i>
<a NAME="E9:21_7_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c15">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">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" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7_1_3"/><i><font color="Green" title="E27">A177</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c15">t</font>
 ;<br/>

<a NAME="E2:21_7_1_3"/><b>then </b><i><font color="Green" title="E28">A178</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">b2</font> = <font color="Maroon" title="c17">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> <b>by </b><i><a class="txt" href="polynom6.html#E1:21_7_1_3"><i><font color="Green" title="E27">A177</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">b</font> = <font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font>;<br/>
<a NAME="E4:21_7_1_3"/><i><font color="Green" title="E29">A179</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:21_7_1_3"/>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1"><i><font color="Green" title="E24">A173</font></i></a>, <a class="txt" href="polynom6.html#E2:21_7_1_3"><i><font color="Green" title="E28">A178</font></i></a></i>;<br/>
<a NAME="E6:21_7_1_3"/><b>then </b><i><font color="Green" title="E30">A180</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c21">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E8:21_7_1_3"/><i><font color="Green" title="E31">A181</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c20">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c21">b2'</font>
 <b>and </b><br/><a NAME="E9:21_7_1_3"/><i><font color="Green" title="E32">A182</font></i>: 
<font color="Maroon" title="c16">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b2'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:21_7_1"><i><font color="Green" title="E26">A176</font></i></a></i>;<br/>
<a NAME="E10:21_7_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c18">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E8:21_7_1_3"><i><font color="Green" title="E31">A181</font></i></a>, <a class="txt" href="polynom6.html#E9:21_7_1_3"><i><font color="Green" title="E32">A182</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E11:21_7_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1_3"><i><font color="Green" title="E29">A179</font></i></a>, <a class="txt" href="polynom6.html#E6:21_7_1_3"><i><font color="Green" title="E30">A180</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">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" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c15">t</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:21_7_1"/><b>then </b>
<font color="Maroon" title="c15">t</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">t''</font> = <font color="Maroon" title="c15">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">t'</font> = <font color="Maroon" title="c15">t</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c17">t''</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c18">t'</font>
;<br/>

<b>thus </b><a NAME="E14:21_7_1"/>
<font color="Maroon" title="c17">t''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t'</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c19">b2</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>;<br/><b>let </b><font color="Maroon" title="c20">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>



<b>assume </b><a NAME="E15:21_7_1"/>
<font color="Maroon" title="c20">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">x</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>
 ;<br/>

<b>hence </b><a NAME="E16:21_7_1"/>
<font color="Maroon" title="c18">t'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_1"><i><font color="Green" title="E24">A173</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span><b> such that </b><br/><a NAME="E5:21_7"/><i><font color="Green" title="E23">A183</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">g</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E3:21_7"><i><font color="Green" title="E22">A171</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">g</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ex <font color="Olive" title="b2">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">b2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> );<br/>
<a NAME="E7:21_7"/><i><font color="Green" title="E24">A184</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c14">x</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c15">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c16">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E2:21_7_2"/><i><font color="Green" title="E25">A185</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c16">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T6" target="_self" title="POLYNOM6:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c15">b1</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c17">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">b1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c16">b2</font>
;<br/>

<b>thus </b><a NAME="E4:21_7_2"/>
<font color="Maroon" title="c14">x</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c16">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_7_2"><i><font color="Green" title="E25">A185</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:21_7_2"/>
<font color="Maroon" title="c17">y</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">kk</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>, <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font><b> such that </b><br/><a NAME="E9:21_7"/><i><font color="Green" title="E25">A186</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c14">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="polynom6.html#E7:21_7"><i><font color="Green" title="E24">A184</font></i></a>);<br/></i>
<a NAME="E10:21_7"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">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" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_7_3"/><i><font color="Green" title="E26">A187</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c13">g</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">b1</font> = <font color="Maroon" title="c15">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font> ;<br/>
<b>consider </b><font color="Maroon" title="c17">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:21_7_3"/><i><font color="Green" title="E27">A188</font></i>: 
<font color="Maroon" title="c17">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b1</font>
 <b>and </b><br/><a NAME="E5:21_7_3"/><i><font color="Green" title="E28">A189</font></i>: 
for <font color="Olive" title="b1">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> holds <br/><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_7"><i><font color="Green" title="E23">A183</font></i></a></i>;<br/>
<a NAME="E6:21_7_3"/>
<font color="Maroon" title="c13">g</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E1:21_7_3"><i><font color="Green" title="E26">A187</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E7:21_7_3"/><b>then </b><i><font color="Green" title="E29">A190</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c17">h</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_3"><i><font color="Green" title="E27">A188</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E9:21_7_3"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c18">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="polynom6.html#E4:21_7_3"><i><font color="Green" title="E27">A188</font></i></a>, <a class="txt" href="polynom6.html#E7:21_7_3"><i><font color="Green" title="E29">A190</font></i></a>, <a class="ref" href="polynom2.html#T19" title="POLYNOM2:th.19">POLYNOM2:19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E11:21_7_3"/><i><font color="Green" title="E30">A191</font></i>: 
<font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c18">h</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>;<br/>
<a NAME="E12:21_7_3"/><i><font color="Green" title="E31">A192</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:21_7_3"/>
<font color="Maroon" title="c18">h</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:21_7_3"><i><font color="Green" title="E30">A191</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E14:21_7_3"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E5:21_7_3"><i><font color="Green" title="E28">A189</font></i></a></i>;<br/>
<a NAME="E15:21_7_3"/><b>then </b><i><font color="Green" title="E32">A193</font></i>: 
<font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">b1'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c22">b2'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><b> such that </b><br/><a NAME="E17:21_7_3"/><i><font color="Green" title="E33">A194</font></i>: 
<font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">b1'</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c22">b2'</font>
 <b>and </b><br/><a NAME="E18:21_7_3"/><i><font color="Green" title="E34">A195</font></i>: 
<font color="Maroon" title="c14">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1'</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E9:21_7"><i><font color="Green" title="E25">A186</font></i></a></i>;<br/>
<a NAME="E19:21_7_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">kk</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E17:21_7_3"><i><font color="Green" title="E33">A194</font></i></a>, <a class="txt" href="polynom6.html#E18:21_7_3"><i><font color="Green" title="E34">A195</font></i></a>, <a class="ref" href="polynom6.html#T7" target="_self" title="POLYNOM6:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E20:21_7_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">kk</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c11">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E12:21_7_3"><i><font color="Green" title="E31">A192</font></i></a>, <a class="txt" href="polynom6.html#E15:21_7_3"><i><font color="Green" title="E32">A193</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:21_7"/><b>then </b><i><font color="Green" title="E26">A196</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c13">g</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E12:21_7"/><b>then </b>
<font color="Maroon" title="c13">g</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c13">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">g'</font> = <font color="Maroon" title="c15">g</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">o1</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="txt" href="polynom6.html#E11:21_7"><i><font color="Green" title="E26">A196</font></i></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/>
<a NAME="E15:21_7"/><i><font color="Green" title="E27">A197</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">o1</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c17">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">b1</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c19">b2</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font>, <font color="Maroon" title="c20">Q1</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">o2</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E2:21_7_4"/><i><font color="Green" title="E28">A198</font></i>: 
<font color="Maroon" title="c20">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font>
 <b>and </b><br/><a NAME="E3:21_7_4"/><i><font color="Green" title="E29">A199</font></i>: 
<font color="Maroon" title="c17">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c19">b2</font>
 <b>and </b><br/><a NAME="E4:21_7_4"/><i><font color="Green" title="E30">A200</font></i>: 
<span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g'</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">b2</font>
 <b>by </b><i><a class="ref" href="polynom6.html#D2" target="_self" title="POLYNOM6:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:21_7_4"/><i><font color="Green" title="E31">A201</font></i>: 
<font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g'</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b1</font>
 <b>and </b><br/><a NAME="E7:21_7_4"/><i><font color="Green" title="E32">A202</font></i>: 
for <font color="Olive" title="b1">b2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">o2</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span>  st <font color="Olive" title="b2">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> holds <br/><font color="Maroon" title="c21">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E5:21_7"><i><font color="Green" title="E23">A183</font></i></a></i>;<br/><b>thus </b><a NAME="E8:21_7_4"/>
<font color="Maroon" title="c11">s</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g'</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E2:21_7_4"><i><font color="Green" title="E28">A198</font></i></a>, <a class="txt" href="polynom6.html#E3:21_7_4"><i><font color="Green" title="E29">A199</font></i></a>, <a class="txt" href="polynom6.html#E4:21_7_4"><i><font color="Green" title="E30">A200</font></i></a>, <a class="txt" href="polynom6.html#E6:21_7_4"><i><font color="Green" title="E31">A201</font></i></a>, <a class="txt" href="polynom6.html#E7:21_7_4"><i><font color="Green" title="E32">A202</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><font color="Maroon" title="c10">y</font> = 
 <a href="polynom6.html#K2" title="POLYNOM6:func.2">Compress</a> <font color="Maroon" title="c16">g'</font>
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">g</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E18:21_7"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E15:21_7"><i><font color="Green" title="E27">A197</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


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


</div>
