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<b>set </b><font color="Maroon" title="c3">PRPNR</font> =  <a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <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>;<br/>
<b>set </b><font color="Maroon" title="c4">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="c5">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c7">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>;<br/><b>assume </b><a NAME="E1:37_4_1_1"/><i><font color="Green" title="E33">A102</font></i>: 
<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">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">y'</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x'</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">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/><b>reconsider </b><font color="Maroon" title="c10">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="c7">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_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">bn'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_4_1_1_1"/>
<font color="Maroon" title="c11">bn'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">bn</font> = <font color="Maroon" title="c11">bn'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c13">xbn</font> = <font color="Maroon" title="c8">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">bn</font>, <font color="Maroon" title="c14">ybn</font> = <font color="Maroon" title="c9">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">bn</font> 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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">b'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_4_1_1_1_1"/>
<font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c15">b'</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="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon" title="c17">bn1</font> = <font color="Maroon" title="c16">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">bn</font>;<br/><a NAME="E3:37_4_1_1_1_1"/><i><font color="Green" title="E34">A103</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">bn</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">bn</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">bn</font> )
 <b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>;<br/><b>then </b><font color="Maroon" title="c13">xbn</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b</font> = 
<font color="Maroon" title="c10">Py</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">bn</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:37_4_1_1"><i><font color="Green" title="E33">A102</font></i></a>, <a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c14">ybn</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c16">b</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:37_4_1_1_1_1"><i><font color="Green" title="E34">A103</font></i></a>, <a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>
;<br/><b>hence </b><a NAME="E5:37_4_1_1_1_1"/>
<font color="Maroon" title="c13">xbn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ybn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b'</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:37_4_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">bn'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">bn'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:37_4_1_1"/>
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_4_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="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a></i>;<br/>


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