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

<b>set </b><font color="Maroon" title="c3">PNR</font> =  <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">P</font> =  <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<b>thus </b><a NAME="E1:37_1_1"/>
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is <a href="grcat_1.html#V1" title="GRCAT_1:attr.1">additive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13" title="GRCAT_1:def.13">GRCAT_1:def 13</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c9">xy'</font> = <font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E2:37_1_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c9">xy'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">x'</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c8">y'</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Pxy</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span>, <font color="Maroon" title="c11">Px</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c12">Py</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1_1"/>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">b'</font> = <font color="Maroon" title="c13">b</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">xbn</font> = <font color="Maroon" title="c7">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span>, <font color="Maroon" title="c16">ybn</font> = <font color="Maroon" title="c8">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:37_1_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">bn</font> = <font color="Maroon" title="c14">b'</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">xybn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c8">y'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E7:37_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="polynom3.html#K8" title="POLYNOM3:func.8">+</a> <font color="Maroon" title="c8">y'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>;<br/><a NAME="E8:37_1_1_1_1"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c18">xybn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">xbn</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c16">ybn</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E9:37_1_1_1_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c15">xbn</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c14">b'</font> &amp; <font color="Maroon" title="c16">ybn</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c14">b'</font> )
 <b>by </b><i><a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>;<br/><b>thus </b><font color="Maroon" title="c10">Pxy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">xbn</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c16">ybn</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c17">bn</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E2:37_1_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="hilbasis.html#E8:37_1_1_1_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Px</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c14">b'</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c14">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E9:37_1_1_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Px</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c12">Py</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> = 
<font color="Maroon" title="c11">Px</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c12">Py</font>
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><br/>


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


</div>
