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<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="vectsp_1.html#V3" title="VECTSP_1:attr.3">strict</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:186_1_2"/><i><font color="Green" title="E101">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="E2:186_1_2"/><i><font color="Green" title="E102">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="rlvect_1.html#K1" title="RLVECT_1: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#K22" title="POLYNOM1:func.22">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E3:186_1_2"/><i><font color="Green" title="E103">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="group_1.html#K1" title="GROUP_1: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#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E4:186_1_2"/><i><font color="Green" title="E104">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#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E5:186_1_2"/><i><font color="Green" title="E105">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#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E6:186_1_2"/><i><font color="Green" title="E106">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="E7:186_1_2"/><i><font color="Green" title="E107">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="rlvect_1.html#K1" title="RLVECT_1: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#K22" title="POLYNOM1:func.22">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E8:186_1_2"/><i><font color="Green" title="E108">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="group_1.html#K1" title="GROUP_1: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#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E9:186_1_2"/><i><font color="Green" title="E109">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#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 <b>and </b><br/><a NAME="E10:186_1_2"/><i><font color="Green" title="E110">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#K27" title="POLYNOM1:func.27">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>
 ;<br/>

<div><i><font color="Green" title="E111">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="186_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="E1:186_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="E2:186_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#E1:186_1_2"><i><font color="Green" title="E101">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E3:186_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#E6:186_1_2"><i><font color="Green" title="E106">A27</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:186_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="E3:186_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#E6:186_1_2"><i><font color="Green" title="E106">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:186_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#E1:186_1_2"><i><font color="Green" title="E101">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:186_1_2"/><b>then </b><i><font color="Green" title="E112">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="186_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#E1:186_1_2"><i><font color="Green" title="E101">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#E11:186_1_2"><i><font color="Green" title="E111">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="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">it1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">a1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">b1</font>

<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c10">q</font>
<b>by </b><i><a class="txt" href="polynom1.html#E2:186_1_2"><i><font color="Green" title="E102">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">a1'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c14">b1'</font>
<b>by </b><i><a class="txt" href="polynom1.html#E7:186_1_2"><i><font color="Green" title="E107">A28</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</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="E14:186_1_2"/><b>then </b><i><font color="Green" title="E113">A34</font></i>: 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">it1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">it2</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E12:186_1_2"><i><font color="Green" title="E112">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="186_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#E1:186_1_2"><i><font color="Green" title="E101">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#E11:186_1_2"><i><font color="Green" title="E111">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="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">it1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">a1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b1</font>

<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c10">q</font>
<b>by </b><i><a class="txt" href="polynom1.html#E3:186_1_2"><i><font color="Green" title="E103">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">a1'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">b1'</font>
<b>by </b><i><a class="txt" href="polynom1.html#E8:186_1_2"><i><font color="Green" title="E108">A29</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</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="E16:186_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#E4:186_1_2"><i><font color="Green" title="E104">A25</font></i></a>, <a class="txt" href="polynom1.html#E5:186_1_2"><i><font color="Green" title="E105">A26</font></i></a>, <a class="txt" href="polynom1.html#E9:186_1_2"><i><font color="Green" title="E109">A30</font></i></a>, <a class="txt" href="polynom1.html#E10:186_1_2"><i><font color="Green" title="E110">A31</font></i></a>, <a class="txt" href="polynom1.html#E12:186_1_2"><i><font color="Green" title="E112">A33</font></i></a>, <a class="txt" href="polynom1.html#E14:186_1_2"><i><font color="Green" title="E113">A34</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


</div>
