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<b>let </b><font color="Maroon">R</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="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> , <font color="Maroon">S</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="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">P</font> is <a href="quofield.html#V4">RingIsomorphism</a> &amp; <font color="Maroon">R</font> is <a href="ideal_1.html#V7">Noetherian</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">S</font>;<br/><b>consider </b><font color="Maroon">PInv</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:33_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">PInv</font> is <a href="quofield.html#V4">RingIsomorphism</a> &amp; <font color="Maroon">PInv</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">P</font> <a href="tops_2.html#NK3" target="_self">"</a>  )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">PI</font> = <font color="Maroon">PInv</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font> as   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font> <b>by </b><i>, </i>;<br/><a NAME="E4:33_1"/>
<font color="Maroon">PI</font> is <a href="ideal_1.html#V6">finitely_generated</a>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D27">IDEAL_1:def 27</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:33_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">PInv</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D26">IDEAL_1:def 26</a></i>;<br/><a NAME="E7:33_1"/>
( <font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a> &amp; <font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a> )
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E8:33_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a>, <a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a></i>;<br/><a NAME="E9:33_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E10:33_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">PInv</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/><a NAME="E11:33_1"/>
<font color="Maroon">P</font> is <a href="quofield.html#V2">RingEpimorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E12:33_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22">QUOFIELD:def 22</a></i>;<br/><a NAME="E13:33_1"/><b>then </b>
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">PInv</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/><a NAME="E14:33_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E15:33_1"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V6">finitely_generated</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
<font color="Maroon">S</font> is <a href="ideal_1.html#V7">Noetherian</a>
 <b>by </b><i><a class="ref" href="ideal_1.html#D27">IDEAL_1:def 27</a></i>;<br/>


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