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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="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#V4" title="GROUP_1:attr.4">associative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c4">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>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">x'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c6">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">lc'</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E5:32_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c6">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">lc'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:32_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="ideal_1.html#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">E</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="ideal_1.html#R1" title="IDEAL_1:pred.1">represents</a> <font color="Maroon" title="c7">lc'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35" title="IDEAL_1:th.35">IDEAL_1:35</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green" title="E33">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">lc'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">lc</font> holds <br/><font color="Maroon" title="c9">lc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:32_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="ideal_1.html#T36" title="IDEAL_1:th.36">IDEAL_1:36</a></i>;<br/><a NAME="E10:32_1"/>
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:32"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E11:32_1"/><b>then </b>
<font color="Maroon" title="c4">P</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/><a NAME="E12:32_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c9">lc</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:32_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="hilbasis.html#E5:32_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="hilbasis.html#E7:32_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="hilbasis.html#E9:32_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="hilbasis.html#T24" target="_self" title="HILBASIS:th.24">Th24</a></i>;<br/><b>hence </b><a NAME="E13:32_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:32"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E3:32_2"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></i>;<br/><a NAME="E4:32_2"/>
( <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a> &amp; <font color="Maroon" title="c4">P</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:32"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E5:32_2"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
( <font color="Maroon" title="c4">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="c4">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="E6:32_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/><a NAME="E7:32_2"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:32_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/><a NAME="E8:32_2"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:32_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E9:32_2"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:32"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="hilbasis.html#T25" target="_self" title="HILBASIS:th.25">Th25</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">E</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E11:32_2"/><i><font color="Green" title="E37">A13</font></i>: 
<font color="Maroon" title="c7">E</font> <a href="ideal_1.html#R1" title="IDEAL_1:pred.1">represents</a> <font color="Maroon" title="c6">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35" title="IDEAL_1:th.35">IDEAL_1:35</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">lc'</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E13:32_2"/><i><font color="Green" title="E38">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">lc'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">lc'</font> holds <br/><font color="Maroon" title="c8">lc'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:32_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="hilbasis.html#E8:32_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="hilbasis.html#E11:32_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="ideal_1.html#T36" title="IDEAL_1:th.36">IDEAL_1:36</a></i>;<br/><a NAME="E14:32_2"/>
<font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:32_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="quofield.html#D24" title="QUOFIELD:def.24">QUOFIELD:def 24</a></i>;<br/><a NAME="E15:32_2"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D23" title="QUOFIELD:def.23">QUOFIELD:def 23</a></i>;<br/><a NAME="E16:32_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">lc'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:32_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="hilbasis.html#E11:32_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="hilbasis.html#E13:32_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="hilbasis.html#T24" target="_self" title="HILBASIS:th.24">Th24</a></i>;<br/><a NAME="E17:32_2"/><b>then </b><i><font color="Green" title="E39">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></i>;<br/><a NAME="E18:32_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">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="c1">R</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:32_2"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E17:32_2"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E20:32_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:32_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="hilbasis.html#E5:32_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="hilbasis.html#E8:32_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:32"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon" title="c4">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">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">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="hilbasis.html#E3:32"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
