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<b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">pq</font> = <font color="Maroon" title="c5">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c8">qp</font> = <font color="Maroon" title="c6">q</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font> as   <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">n</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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> <font color="Maroon" title="c1">n</font>;<br/><b>consider </b><font color="Maroon" title="c10">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="c2">L</font><b> such that </b><br/><a NAME="E2:174_1_1_1"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c7">pq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">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="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E3:174_1_1_1"/><i><font color="Green" title="E100">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c9">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:174_1_1_1"/><i><font color="Green" title="E101">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="c10">s</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</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">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c10">s</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="c5">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" target="_self" title="POLYNOM1:def.26">Def26</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">t</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="c2">L</font><b> such that </b><br/><a NAME="E6:174_1_1_1"/><i><font color="Green" title="E102">A4</font></i>: 
<font color="Maroon" title="c8">qp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">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="Maroon" title="c11">t</font>
 <b>and </b><br/><a NAME="E7:174_1_1_1"/><i><font color="Green" title="E103">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">t</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="c9">b</font></span>)</span>
 <b>and </b><br/><a NAME="E8:174_1_1_1"/><i><font color="Green" title="E104">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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="c11">t</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</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">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c11">t</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="c6">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" target="_self" title="POLYNOM1:def.26">Def26</a></i>;<br/><a NAME="E9:174_1_1_1"/><i><font color="Green" title="E105">A7</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s</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="c9">b</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">t</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="c9">b</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">ds</font> =  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<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> ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1: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">b1</font>,<font color="Olive" title="b2">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <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">b2</font>,<font color="Olive" title="b1">b1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> );<br/><div><i><font color="Green" title="E106">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:174_1_1_1_1"/><i><font color="Green" title="E107">A9</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ds</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c15">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:174_1_1_1_1"/><i><font color="Green" title="E108">A10</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c13">e</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="c14">b1</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:174_1_1_1_1"/><i><font color="Green" title="E109">A11</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c14">b1</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c15">b2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="ref" href="polynom1.html#T72" target="_self" title="POLYNOM1:th.72">Th72</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:174_1_1_1_1"/><i><font color="Green" title="E110">A12</font></i>: 
<font color="Maroon" title="c16">d</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="c9">b</font></span>)</span>
 <b>and </b><br/><a NAME="E7:174_1_1_1_1"/><i><font color="Green" title="E111">A13</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c16">d</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="c15">b2</font>,<font color="Maroon" title="c14">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="polynom1.html#E4:174_1_1_1_1"><i><font color="Green" title="E109">A11</font></i></a>, <a class="ref" href="polynom1.html#T73" target="_self" title="POLYNOM1:th.73">Th73</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">d</font> = <font color="Maroon" title="c16">d</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c18">d</font> = <font color="Maroon" title="c17">d</font>;<br/><b>thus </b><a NAME="E9:174_1_1_1_1"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ds</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="polynom1.html#E6:174_1_1_1_1"><i><font color="Green" title="E110">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>thus </b><a NAME="E10:174_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c18">d</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c14">b1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">b2</font>
;<br/>

<b>thus </b><a NAME="E1:174_1_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="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e</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="c14">b1</font>,<font color="Maroon" title="c15">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</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="c15">b2</font>,<font color="Maroon" title="c14">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="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_1"><i><font color="Green" title="E107">A9</font></i></a>, <a class="txt" href="polynom1.html#E3:174_1_1_1_1"><i><font color="Green" title="E108">A10</font></i></a>, <a class="txt" href="polynom1.html#E6:174_1_1_1_1"><i><font color="Green" title="E110">A12</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1_1"><i><font color="Green" title="E111">A13</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></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c13">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c12">ds</font>,<font color="Maroon" title="c12">ds</font><b> such that </b><br/><a NAME="E13:174_1_1_1"/><i><font color="Green" title="E107">A14</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ds</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="polynom1.html#E11:174_1_1_1"><i><font color="Green" title="E106">A8</font></i></a>);<br/></i><a NAME="E14:174_1_1_1"/><i><font color="Green" title="E108">A15</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">ds</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">ds</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E15:174_1_1_1"/><i><font color="Green" title="E109">A16</font></i>: 
<font color="Maroon" title="c13">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="174_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x1</font>, <font color="Maroon" title="c15">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><b>that </b><br/><a NAME="E1:174_1_1_1_2"/><i><font color="Green" title="E110">A17</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font>
 <b>and </b><br/><a NAME="E2:174_1_1_1_2"/><i><font color="Green" title="E111">A18</font></i>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">f</font>
 <b>and </b><br/><a NAME="E3:174_1_1_1_2"/><i><font color="Green" title="E112">A19</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c16">b1</font>, <font color="Maroon" title="c17">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:174_1_1_1_2"/><i><font color="Green" title="E113">A20</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</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="c16">b1</font>,<font color="Maroon" title="c17">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font></span>)</span> <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="c17">b2</font>,<font color="Maroon" title="c16">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="polynom1.html#E13:174_1_1_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_2"><i><font color="Green" title="E110">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">b3</font>, <font color="Maroon" title="c19">b4</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:174_1_1_1_2"/><i><font color="Green" title="E114">A21</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</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="c18">b3</font>,<font color="Maroon" title="c19">b4</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x2</font></span>)</span> <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="c19">b4</font>,<font color="Maroon" title="c18">b3</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E13:174_1_1_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_2"><i><font color="Green" title="E111">A18</font></i></a></i>;<br/>
<a NAME="E8:174_1_1_1_2"/>
( <font color="Maroon" title="c17">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">b4</font> &amp; <font color="Maroon" title="c16">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">b3</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1_1_2"><i><font color="Green" title="E112">A19</font></i></a>, <a class="txt" href="polynom1.html#E5:174_1_1_1_2"><i><font color="Green" title="E113">A20</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1_2"><i><font color="Green" title="E114">A21</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E9:174_1_1_1_2"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_2"><i><font color="Green" title="E110">A17</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_2"><i><font color="Green" title="E111">A18</font></i></a>, <a class="txt" href="polynom1.html#E5:174_1_1_1_2"><i><font color="Green" title="E113">A20</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1_2"><i><font color="Green" title="E114">A21</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:174_1_1_1"/>
<font color="Maroon" title="c12">ds</font> <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="c13">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">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="E1:174_1_1_1_3"/><i><font color="Green" title="E110">A22</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ds</font>
 ;<br/>

<a NAME="E2:174_1_1_1_3"/><b>then </b><i><font color="Green" title="E111">A23</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c13">f</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:174_1_1_1_3"/><b>then </b><i><font color="Green" title="E112">A24</font></i>: 
<font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <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="c13">f</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:174_1_1_1_3"/><i><font color="Green" title="E113">A25</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</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="c15">b1</font>,<font color="Maroon" title="c16">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <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="c16">b2</font>,<font color="Maroon" title="c15">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="polynom1.html#E13:174_1_1_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_3"><i><font color="Green" title="E110">A22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">b3</font>, <font color="Maroon" title="c18">b4</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:174_1_1_1_3"/><i><font color="Green" title="E114">A26</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <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="c17">b3</font>,<font color="Maroon" title="c18">b4</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <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="c18">b4</font>,<font color="Maroon" title="c17">b3</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E13:174_1_1_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_3"><i><font color="Green" title="E111">A23</font></i></a></i>;<br/>
<a NAME="E8:174_1_1_1_3"/>
( <font color="Maroon" title="c17">b3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">b2</font> &amp; <font color="Maroon" title="c18">b4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c15">b1</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:174_1_1_1_3"><i><font color="Green" title="E113">A25</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1_3"><i><font color="Green" title="E114">A26</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E9:174_1_1_1_3"/>
<font color="Maroon" title="c14">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="c13">f</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_3"><i><font color="Green" title="E110">A22</font></i></a>, <a class="txt" href="polynom1.html#E3:174_1_1_1_3"><i><font color="Green" title="E112">A24</font></i></a>, <a class="txt" href="polynom1.html#E5:174_1_1_1_3"><i><font color="Green" title="E113">A25</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1_3"><i><font color="Green" title="E114">A26</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:174_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">ds</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c13">f</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s</font> <b>by </b><i><a class="txt" href="polynom1.html#E15:174_1_1_1"><i><font color="Green" title="E109">A16</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:174_1_1_1_4"/><i><font color="Green" title="E110">A27</font></i>: 
<font color="Maroon" title="c15">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="c11">t</font>
 ;<br/><a NAME="E2:174_1_1_1_4"/><b>then </b><i><font color="Green" title="E111">A28</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</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="c14">f</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:174_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c14">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">ds</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">fi</font> = <font color="Maroon" title="c14">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>consider </b><font color="Maroon" title="c17">b1</font>, <font color="Maroon" title="c18">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:174_1_1_1_4"/><i><font color="Green" title="E112">A29</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c15">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="c17">b1</font>,<font color="Maroon" title="c18">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c11">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">b1</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c18">b2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E8:174_1_1_1"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_4"><i><font color="Green" title="E110">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">b3</font>, <font color="Maroon" title="c20">b4</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E8:174_1_1_1_4"/><i><font color="Green" title="E113">A30</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c16">fi</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="c19">b3</font>,<font color="Maroon" title="c20">b4</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c10">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c19">b3</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c20">b4</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E4:174_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_4"><i><font color="Green" title="E111">A28</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">b5</font>, <font color="Maroon" title="c22">b6</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E10:174_1_1_1_4"/><i><font color="Green" title="E114">A31</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c21">b5</font>,<font color="Maroon" title="c22">b6</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font></span>)</span> <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="c22">b6</font>,<font color="Maroon" title="c21">b5</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="polynom1.html#E9:174_1_1_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="polynom1.html#E13:174_1_1_1"><i><font color="Green" title="E107">A14</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_4"><i><font color="Green" title="E110">A27</font></i></a></i>;<br/><a NAME="E11:174_1_1_1_4"/><i><font color="Green" title="E115">A32</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s</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="c11">t</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E12:174_1_1_1_4"/><i><font color="Green" title="E116">A33</font></i>: 
( <font color="Maroon" title="c11">t</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c10">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">fi</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_4"><i><font color="Green" title="E110">A27</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_4"><i><font color="Green" title="E111">A28</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="E13:174_1_1_1_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">s</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="c9">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E3:174_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:174_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c15">i</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="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c16">fi</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="c9">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">fi</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E14:174_1_1_1"><i><font color="Green" title="E108">A15</font></i></a>, <a class="txt" href="polynom1.html#E1:174_1_1_1_4"><i><font color="Green" title="E110">A27</font></i></a>, <a class="txt" href="polynom1.html#E2:174_1_1_1_4"><i><font color="Green" title="E111">A28</font></i></a>, <a class="txt" href="polynom1.html#E11:174_1_1_1_4"><i><font color="Green" title="E115">A32</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="E15:174_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c17">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">b5</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b6</font> &amp; <font color="Maroon" title="c19">b3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">b6</font> &amp; <font color="Maroon" title="c20">b4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c21">b5</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E6:174_1_1_1_4"><i><font color="Green" title="E112">A29</font></i></a>, <a class="txt" href="polynom1.html#E8:174_1_1_1_4"><i><font color="Green" title="E113">A30</font></i></a>, <a class="txt" href="polynom1.html#E10:174_1_1_1_4"><i><font color="Green" title="E114">A31</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/><b>hence </b><a NAME="E16:174_1_1_1_4"/>
<font color="Maroon" title="c11">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:174_1_1_1_4"><i><font color="Green" title="E112">A29</font></i></a>, <a class="txt" href="polynom1.html#E8:174_1_1_1_4"><i><font color="Green" title="E113">A30</font></i></a>, <a class="txt" href="polynom1.html#E12:174_1_1_1_4"><i><font color="Green" title="E116">A33</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:174_1_1_1"/>
<font color="Maroon" title="c7">pq</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">qp</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:174_1_1_1"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="polynom1.html#E3:174_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="polynom1.html#E6:174_1_1_1"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="polynom1.html#E7:174_1_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="ref" href="rlvect_2.html#T8" title="RLVECT_2:th.8">RLVECT_2:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:174_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">q</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</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>
