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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">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<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="E1:32"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P</font> is <a href="quofield.html#V4">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon">lc'</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E5:32_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc'</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/><b>consider </b><font color="Maroon">E</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">E</font> <a href="ideal_1.html#R1">represents</a> <font color="Maroon">lc'</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35">IDEAL_1:35</a></i>;<br/><b>consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E9:32_1"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lc</font> holds <br/><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T36">IDEAL_1:36</a></i>;<br/><a NAME="E10:32_1"/>
<font color="Maroon">P</font> is <a href="quofield.html#V3">RingMonomorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E11:32_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a></i>;<br/><a NAME="E12:32_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E13:32_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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><a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> <a href="tarski.html#R1">c=</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><a class="ref" href="tarski.html#D3">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> 
 ;<br/><b>then consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E3:32_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/><a NAME="E4:32_2"/>
( <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="E5:32_2"/><b>then </b><i><font color="Green">E53</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="E6:32_2"/><b>then </b><i><font color="Green">E54</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> )
 ;<br/><a NAME="E7:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</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="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="relat_1.html#K10">"</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>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/><a NAME="E8:32_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</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="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/><a NAME="E9:32_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 <b>by </b><i>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<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="E11:32_2"/><i><font color="Green">E59</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>consider </b><font color="Maroon">E</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="zfmisc_1.html#K3">:]</a></span><b> such that </b><br/><a NAME="E13:32_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">E</font> <a href="ideal_1.html#R1">represents</a> <font color="Maroon">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35">IDEAL_1:35</a></i>;<br/><b>consider </b><font color="Maroon">lc'</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E15:32_2"/><i><font color="Green">E76</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc'</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lc'</font> holds <br/><font color="Maroon">lc'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">PInv</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">PInv</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">PInv</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">E</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K7">`3</a> </span>)</span></span>)</span> ) )
 <b>by </b><i>, , , <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="ideal_1.html#T36">IDEAL_1:36</a></i>;<br/><a NAME="E16:32_2"/>
<font color="Maroon">PInv</font> is <a href="quofield.html#V3">RingMonomorphism</a>
 <b>by </b><i>, <a class="ref" href="quofield.html#D24">QUOFIELD:def 24</a></i>;<br/><a NAME="E17:32_2"/><b>then </b>
<font color="Maroon">PInv</font> is <a href="quofield.html#V1">RingHomomorphism</a>
 <b>by </b><i><a class="ref" href="quofield.html#D23">QUOFIELD:def 23</a></i>;<br/><a NAME="E18:32_2"/><b>then </b>
<font color="Maroon">PInv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc'</font>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, , </i>;<br/><a NAME="E19:32_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">PInv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</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#T60">IDEAL_1:60</a></i>;<br/><a NAME="E20:32_2"/>
 <a href="relset_1.html#K4">dom</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">R</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:32_2"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">PInv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E22:32_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T57">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">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> <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>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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