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<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a> holds <br/><font color="Maroon" title="c2">S</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a></span><br/><b>let </b><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a> implies <font color="Maroon" title="c2">S</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a> )</span><br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a></span><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" title="c4">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">finitely_generated</a></span><br/><b>reconsider </b><font color="Maroon" title="c5">PI</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">I</font> as   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="hilbasis.html#T22" target="_self" title="HILBASIS:th.22">Th22</a>, <a class="ref" href="hilbasis.html#T25" target="_self" title="HILBASIS:th.25">Th25</a></i>;<br/><a NAME="E2:33_1"/>
<font color="Maroon" title="c5">PI</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">finitely_generated</a>
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="ideal_1.html#D27" title="IDEAL_1:def.27">IDEAL_1:def 27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">F</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green" title="E31">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D26" title="IDEAL_1:def.26">IDEAL_1:def 26</a></i>;<br/><a NAME="E5:33_1"/>
( <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <font color="Maroon" title="c3">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> )
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E6:33_1"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c3">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a>, <a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/><a NAME="E7:33_1"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E8:33_1"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:33_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/><a NAME="E9:33_1"/>
<font color="Maroon" title="c3">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E10:33_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="quofield.html#D22" title="QUOFIELD:def.22">QUOFIELD:def 22</a></i>;<br/><a NAME="E11:33_1"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="#E8:33_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/><a NAME="E12:33_1"/><b>then </b>
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E4:33_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="hilbasis.html#T26" target="_self" title="HILBASIS:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E13:33_1"/>
<font color="Maroon" title="c4">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">finitely_generated</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
<font color="Maroon" title="c2">S</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a>
 <b>by </b><i><a class="ref" href="ideal_1.html#D27" title="IDEAL_1:def.27">IDEAL_1:def 27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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