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

<b>let </b><font color="Maroon" title="c3">it1</font>, <font color="Maroon" title="c4">it2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:179_1_2"/><i><font color="Green" title="E100">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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">it1</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )
 <b>and </b><br/><a NAME="E2:179_1_2"/><i><font color="Green" title="E101">A23</font></i>: 
for <font color="Olive" title="b1">x</font>, <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">it1</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K21" title="POLYNOM1:func.21">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E3:179_1_2"/><i><font color="Green" title="E102">A24</font></i>: 
for <font color="Olive" title="b1">x</font>, <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">it1</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E4:179_1_2"/><i><font color="Green" title="E103">A25</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E5:179_1_2"/><i><font color="Green" title="E104">A26</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E6:179_1_2"/><i><font color="Green" title="E105">A27</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</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="c4">it2</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )
 <b>and </b><br/><a NAME="E7:179_1_2"/><i><font color="Green" title="E106">A28</font></i>: 
for <font color="Olive" title="b1">x</font>, <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="c4">it2</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K21" title="POLYNOM1:func.21">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E8:179_1_2"/><i><font color="Green" title="E107">A29</font></i>: 
for <font color="Olive" title="b1">x</font>, <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="c4">it2</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E9:179_1_2"/><i><font color="Green" title="E108">A30</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E10:179_1_2"/><i><font color="Green" title="E109">A31</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 ;<br/>

<div><i><font color="Green" title="E110">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:179_1_2_1_1"/>
<font color="Maroon" title="c5">x</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">it1</font>
 ;<br/><a NAME="E2:179_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:179_1_2"><i><font color="Green" title="E100">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:179_1_2_1_1"/>
<font color="Maroon" title="c5">x</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="c4">it2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:179_1_2"><i><font color="Green" title="E105">A27</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:179_1_2_1"/>
<font color="Maroon" title="c5">x</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="c4">it2</font>
 ;<br/><a NAME="E3:179_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E6:179_1_2"><i><font color="Green" title="E105">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:179_1_2_1"/>
<font color="Maroon" title="c5">x</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">it1</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:179_1_2"><i><font color="Green" title="E100">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:179_1_2"/><b>then </b><i><font color="Green" title="E111">A33</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">it1</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 <font color="Maroon" title="c4">it2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c6">b</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="polynom1.html#E1:179_1_2"><i><font color="Green" title="E100">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> <b>by </b><i><a class="txt" href="polynom1.html#E11:179_1_2"><i><font color="Green" title="E110">A32</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">a1'</font> = <font color="Maroon" title="c11">a'</font>, <font color="Maroon" title="c14">b1'</font> = <font color="Maroon" title="c12">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> ;<br/><b>thus </b>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">it1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">a1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">b1</font>

<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="polynom1.html#K21" title="POLYNOM1:func.21">+</a> <font color="Maroon" title="c10">q</font>
<b>by </b><i><a class="txt" href="polynom1.html#E2:179_1_2"><i><font color="Green" title="E101">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">a1'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c14">b1'</font>
<b>by </b><i><a class="txt" href="polynom1.html#E7:179_1_2"><i><font color="Green" title="E106">A28</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">it2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>

;<br/></div><b>end;</b></div>
<a NAME="E14:179_1_2"/><b>then </b><i><font color="Green" title="E112">A34</font></i>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">it1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">it2</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E12:179_1_2"><i><font color="Green" title="E111">A33</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c6">b</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="polynom1.html#E1:179_1_2"><i><font color="Green" title="E100">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> <b>by </b><i><a class="txt" href="polynom1.html#E11:179_1_2"><i><font color="Green" title="E110">A32</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">a1'</font> = <font color="Maroon" title="c11">a'</font>, <font color="Maroon" title="c14">b1'</font> = <font color="Maroon" title="c12">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> ;<br/><b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">it1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b1</font>

<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c10">q</font>
<b>by </b><i><a class="txt" href="polynom1.html#E3:179_1_2"><i><font color="Green" title="E102">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">a1'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1'</font>
<b>by </b><i><a class="txt" href="polynom1.html#E8:179_1_2"><i><font color="Green" title="E107">A29</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">it2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:179_1_2"/>
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E4:179_1_2"><i><font color="Green" title="E103">A25</font></i></a>, <a class="txt" href="polynom1.html#E5:179_1_2"><i><font color="Green" title="E104">A26</font></i></a>, <a class="txt" href="polynom1.html#E9:179_1_2"><i><font color="Green" title="E108">A30</font></i></a>, <a class="txt" href="polynom1.html#E10:179_1_2"><i><font color="Green" title="E109">A31</font></i></a>, <a class="txt" href="polynom1.html#E12:179_1_2"><i><font color="Green" title="E111">A33</font></i></a>, <a class="txt" href="polynom1.html#E14:179_1_2"><i><font color="Green" title="E112">A34</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


</div>
