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

<b>set </b><font color="Maroon">PRPNR</font> =  <a href="polynom3.html#K16">Polynom-Ring</a> <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>;<br/>
<b>set </b><font color="Maroon">PN1R</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <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>;<br/>
<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">CPN1R</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>;<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>reconsider </b><font color="Maroon">p1</font> =  <a href="group_1.html#K2">1.</a> <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> 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/>
<b>reconsider </b><font color="Maroon">p1'</font> =  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span> 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/>
<a NAME="E3:37_3_1"/>
<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"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <span class="p3">(<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></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">p3</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_1.html#K1">.</a> <font color="Maroon">p1</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_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_3_1_1"/>
<font color="Maroon">x</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">x</font> as    <a href="polynom1.html#M1">Element</a> of  <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>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">p1</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/><i><font color="Green">E276</font></i>: <font color="Maroon">p1</font> = 
 <a href="polynom3.html#K13">1_.</a> <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#T37">POLYNOM3:37</a></i>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <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>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/><a NAME="E5:37_3_1_1"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">p3</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_1_1_2_1"/>
( ( <font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) or <font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> or <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:37_3_1_1_2_1_1"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E279</font></i>: <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hilbasis.html#K1" target="_self">bag_extend</a> 0
<b>by </b><i/>
<br/>.= 
 <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/><font color="Maroon">p2</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</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> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i>, , <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <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#T31">POLYNOM3:31</a></i>
<br/>.= 
 <a href="polynom1.html#K27">1_</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><b>hence </b><font color="Maroon">p3</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">R</font>
<b>by </b><i>, , <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:37_3_1_1_2_1_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> or <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:37_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font></span>)</span> <a href="hilbasis.html#K1" target="_self">bag_extend</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:37_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E281</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_1_2_1_2_1"><b>now </b></a><div class="add"><a NAME="E1:37_3_1_1_2_1_2_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</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> <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>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E2:37_3_1_1_2_1_2_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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:37_3_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E283</font></i>: 
<font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> is   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_1_1_2_1_2_1_1"/>
( <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_3_1_1_2_1_2_1_1_1"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b><font color="Maroon">p2</font> = 
 <a href="group_1.html#K2">1.</a> <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#T31">POLYNOM3:31</a></i>
<br/>.= 
 <a href="polynom1.html#K27">1_</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><b>hence </b><a NAME="E3:37_3_1_1_2_1_2_1_1_1"/>
<font color="Maroon">p3</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, , <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_3_1_1_2_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then </b><font color="Maroon">p2</font> = 
 <a href="rlvect_1.html#K1">0.</a> <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#T31">POLYNOM3:31</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <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/><b>hence </b><a NAME="E3:37_3_1_1_2_1_2_1_1_2"/>
<font color="Maroon">p3</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><font color="Maroon">p3</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K27">1_</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#T84">POLYNOM1:84</a></i>
<br/>.= 
<font color="Maroon">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#T90">POLYNOM1:90</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_3_1_1"/>
<font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:37_3_1"/><b>then </b>
<font color="Maroon">p1'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E7:37_3_1"/>
 <a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font> is <a href="endalg.html#V1">unity-preserving</a>
 <b>by </b><i><a class="ref" href="endalg.html#D8">ENDALG:def 8</a></i>;<br/>


</div>
