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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><a NAME="E1:33_1"/>
<font color="Maroon" title="c3">P</font> is  <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></span>;<br/><a NAME="E2:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D19" title="QUOFIELD:def.19">QUOFIELD:def 19</a></span>;<br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><a NAME="E3:33_1"/>
<font color="Maroon" title="c3">P</font> is  <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="quofield.html#D21" title="QUOFIELD:def.21">QUOFIELD:def 21</a></span>;<br/><a NAME="E4:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D20" title="QUOFIELD:def.20">QUOFIELD:def 20</a></span>;<br/><span class="kw">reconsider </span><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#K7" title="RELSET_1:func.7">.:</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></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></span>;<br/><a NAME="E6:33_1"/>
<font color="Maroon" title="c5">PI</font> is  <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">finitely_generated</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="ideal_1.html#D26" title="IDEAL_1:def.26">IDEAL_1:def 26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:33_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<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#K7" title="RELSET_1:func.7">.:</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D25" title="IDEAL_1:def.25">IDEAL_1:def 25</a></span>;<br/><a NAME="E9:33_1"/>
<font color="Maroon" title="c3">P</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><a NAME="E10:33_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></span>;<br/><a NAME="E11:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K7" title="RELSET_1:func.7">.:</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#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="funct_1.html#T85" title="FUNCT_1:th.85">FUNCT_1:85</a></span>;<br/><a NAME="E12:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>;<br/><a NAME="E13:33_1"/><span class="kw">then </span>
<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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="hilbasis.html#T26" target="_self" title="HILBASIS:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E14:33_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:33"/>
<font color="Maroon" title="c2">S</font> is  <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D26" title="IDEAL_1:def.26">IDEAL_1:def 26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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