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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V7" title="IDEAL_1:attr.7">Noetherian</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">P</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> 0,<font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 <b>by </b><i><a class="ref" href="hilbasis.html#T28" target="_self" title="HILBASIS:th.28">Th28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> $1,<font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V7" title="IDEAL_1:attr.7">Noetherian</a>;<br/>
<a NAME="E4:42"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="hilbasis.html#E1:42"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="hilbasis.html#E3:42"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="hilbasis.html#T27" target="_self" title="HILBASIS:th.27">Th27</a></i>;<br/>
<div><i><font color="Green" title="E41">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c4">P</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">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="c3">k</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 <b>by </b><i><a class="ref" href="hilbasis.html#T31" target="_self" title="HILBASIS:th.31">Th31</a></i>;<br/><a NAME="E4:42_1"/>
 <a href="polynom3.html#K16" title="POLYNOM3:func.16">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="c3">k</font>,<font color="Maroon" title="c1">R</font></span>)</span> is <a href="ideal_1.html#V7" title="IDEAL_1:attr.7">Noetherian</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:42_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="hilbasis.html#E39"><i><font color="Green" title="E36">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:42_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="hilbasis.html#T27" target="_self" title="HILBASIS:th.27">Th27</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E6:42"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="hilbasis.html#E4:42"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="hilbasis.html#E5:42"><i><font color="Green" title="E41">A4</font></i></a>);</i><br/>


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