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<b>let </b><font color="Maroon" title="c1">R</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#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</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="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>





<b>assume </b><a NAME="E1:96"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">s</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:96"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c3">a</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="c4">s</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ;<br/><a NAME="E2:96_1"/><b>then </b>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b1">r</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T64" target="_self" title="IDEAL_1:th.64">Th64</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:96_1"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">r</font>
 ;<br/><b>set </b><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font>;<br/><a NAME="E5:96_1"/><i><font color="Green" title="E51">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></i>;<br/><a NAME="E6:96_1"/>
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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b4">a</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_1"/><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c11">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:96_1_1"/><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">v</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:96_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">v</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">r</font>
<b>by </b><i><a class="txt" href="ideal_1.html#E5:96_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="ideal_1.html#E1:96_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="ideal_1.html#E3:96_1_1"><i><font color="Green" title="E53">A5</font></i></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c6">r</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:96_1_1"/>
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font> st <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b2">v</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:96_1"/><b>then </b><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font> is    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/><a NAME="E8:96_1"/>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">*</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:96"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="ideal_1.html#E4:96_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="binom.html#T5" title="BINOM:th.5">BINOM:5</a></i>;<br/><b>hence </b><a NAME="E9:96_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:96_1"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:96"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="c2">A</font> <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>
