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<b>set </b><font color="Maroon">PNR</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>
<b>thus </b><a NAME="E1:37_1_1"/>
 <a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font> is <a href="grcat_1.html#V1">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="grcat_1.html#D13">GRCAT_1:def 13</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">xy'</font> = <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E2:37_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">xy'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">x'</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Pxy</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span>, <font color="Maroon">Px</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">Py</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">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">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">xbn</font> = <font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">ybn</font> = <font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:37_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">bn</font> = <font color="Maroon">b'</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xybn</font> = <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E7:37_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D6">POLYNOM3:def 6</a></i>;<br/><a NAME="E8:37_1_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">xybn</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">xbn</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">ybn</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E9:37_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">xbn</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">ybn</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font> )
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">Pxy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">xbn</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">ybn</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">Py</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21">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" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<font color="Maroon">Px</font> <a href="polynom1.html#K23">+</a> <font color="Maroon">Py</font>
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><br/>


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


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