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

<b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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="E1:66_1_2"/><i><font color="Green" title="E41">A19</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="c2">P1</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> )
 <b>and </b><br/><a NAME="E2:66_1_2"/><i><font color="Green" title="E42">A20</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="c2">P1</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">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#K2" title="RLVECT_1:func.2">+</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="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E3:66_1_2"/><i><font color="Green" title="E43">A21</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="c2">P1</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">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="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E4:66_1_2"/><i><font color="Green" title="E44">A22</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E5:66_1_2"/><i><font color="Green" title="E45">A23</font></i>: 
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E6:66_1_2"/><i><font color="Green" title="E46">A24</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">P2</font> iff <font color="Olive" title="b1">x</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> )
 <b>and </b><br/><a NAME="E7:66_1_2"/><i><font color="Green" title="E47">A25</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">P2</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">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#K2" title="RLVECT_1:func.2">+</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="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E8:66_1_2"/><i><font color="Green" title="E48">A26</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">P2</font><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">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="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E9:66_1_2"/><i><font color="Green" title="E49">A27</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E10:66_1_2"/><i><font color="Green" title="E50">A28</font></i>: 
 <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <font color="Maroon" title="c3">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<div><i><font color="Green" title="E51">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:66_1_2_1"/>
( <font color="Maroon" title="c4">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="c2">P1</font> iff <font color="Maroon" title="c4">x</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E2:66_1_2_1"/>
( <font color="Maroon" title="c4">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="c2">P1</font> iff <font color="Maroon" title="c4">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">P2</font> )
 <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green" title="E46">A24</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:66_1_2"/><b>then </b><i><font color="Green" title="E52">A30</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P1</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="c3">P2</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="66_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P1</font>;<br/><b>let </b><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P2</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P2</font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P1</font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c5">y</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green" title="E46">A24</font></i></a></i>;<br/><b>thus </b>the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">P1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c4">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c7">y1</font>

<br/>.= 
<font color="Maroon" title="c8">p</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="txt" href="polynom3.html#E2:66_1_2"><i><font color="Green" title="E42">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="polynom3.html#E7:66_1_2"><i><font color="Green" title="E47">A25</font></i></a></i>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">P2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E14:66_1_2"/><b>then </b><i><font color="Green" title="E53">A31</font></i>: 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">P1</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="c3">P2</font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E12:66_1_2"><i><font color="Green" title="E52">A30</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>
<div><i><font color="Green" title="E54">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P1</font>;<br/><b>let </b><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P2</font>;<br/><b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P2</font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P1</font> <b>by </b><i><a class="txt" href="polynom3.html#E11:66_1_2"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom3.html#E1:66_1_2"><i><font color="Green" title="E41">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c5">y</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom3.html#E6:66_1_2"><i><font color="Green" title="E46">A24</font></i></a></i>;<br/><b>thus </b>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c2">P1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y1</font>

<br/>.= 
<font color="Maroon" title="c8">p</font> <a href="polynom3.html#K14" title="POLYNOM3:func.14">*'</a> <font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="txt" href="polynom3.html#E3:66_1_2"><i><font color="Green" title="E43">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="polynom3.html#E8:66_1_2"><i><font color="Green" title="E48">A26</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">P2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>

;<br/></div><b>end;</b></div>
<a NAME="E16:66_1_2"/><i><font color="Green" title="E55">A33</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">unity</a> of <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E5:66_1_2"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="polynom3.html#E10:66_1_2"><i><font color="Green" title="E50">A28</font></i></a></i>;<br/>
<a NAME="E17:66_1_2"/>
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c3">P2</font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E4:66_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="polynom3.html#E9:66_1_2"><i><font color="Green" title="E49">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:66_1_2"/>
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E12:66_1_2"><i><font color="Green" title="E52">A30</font></i></a>, <a class="txt" href="polynom3.html#E14:66_1_2"><i><font color="Green" title="E53">A31</font></i></a>, <a class="txt" href="polynom3.html#E15:66_1_2"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="polynom3.html#E16:66_1_2"><i><font color="Green" title="E55">A33</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


</div>
