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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="ideal_1.html#V7">Noetherian</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> 0,<font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="quofield.html#V4">RingIsomorphism</a>
 <b>by </b><i><a class="ref" href="hilbasis.html#T28" target="_self">Th28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="ideal_1.html#V7">Noetherian</a>;<br/>
<a NAME="E4:42"/><i><font color="Green">E40</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">E38</font></i></a>, <a class="txt" href="hilbasis.html#E3:42"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hilbasis.html#T27" target="_self">Th27</a></i>;<br/>
<div><i><font color="Green">E41</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM7">Function</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="quofield.html#V4">RingIsomorphism</a>
 <b>by </b><i><a class="ref" href="hilbasis.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E4:42_1"/>
 <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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="ideal_1.html#V7">Noetherian</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:42_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hilbasis.html#E39"><i><font color="Green">Lemma36</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:42_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="hilbasis.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E6:42"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="hilbasis.html#E4:42"><i><font color="Green">E40</font></i></a>, <a class="txt" href="hilbasis.html#E5:42"><i><font color="Green">E41</font></i></a>);</i><br/>


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