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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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="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="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/><b>let </b><font color="Maroon" title="c6">P</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 <span class="p1">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c6">P</font>;<br/><b>let </b><font color="Maroon" title="c9">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>















<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="groeb_2.html#R3" title="GROEB_2:pred.3">is_Standard_Representation_of</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">A</font> )
 ;<br/>

<a NAME="E2:58"/><b>then </b><i><font color="Green" title="E30">A2</font></i>: 
(  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c7">A</font> holds <br/> ex <font color="Olive" title="b2">m</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b3">p</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c7">A</font> <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> <font color="Olive" title="b2">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b3">p</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b3">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font> ) ) )
 
<b>by </b><i><a class="ref" href="groeb_2.html#D7" target="_self" title="GROEB_2:def.7">Def7</a></i>;<br/>
<a NAME="E3:58"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/>
<a NAME="E4:58"/><b>then </b>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">B</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">B</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">B</font>
<b>by </b><i><a class="ref" href="algstr_1.html#D5" title="ALGSTR_1:def.5">ALGSTR_1:def 5</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E31">A3</font></i>:  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c8">B</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c5">g</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="groeb_2.html#E2:58"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="groeb_2.html#E28"><i><font color="Green" title="E17">Lm3</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:58_3"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c11">j</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">B</font>
 ;<br/><a NAME="E2:58_3"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">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="c7">A</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/><i><font color="Green" title="E34">A6</font></i>: <font color="Maroon" title="c8">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">j</font> = 
<font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:58_3"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="groeb_2.html#E1:58_3"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c7">A</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">i</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E2:58_3"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>consider </b><font color="Maroon" title="c12">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>, <font color="Maroon" title="c13">p</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E5:58_3"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c7">A</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">p</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c13">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:58"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="groeb_2.html#E2:58_3"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="groeb_2.html#D7" target="_self" title="GROEB_2:def.7">Def7</a></i>;<br/><b>thus </b><a NAME="E6:58_3"/>
 ex <font color="Olive" title="b1">m</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ex <font color="Olive" title="b2">p</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c8">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">p</font> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:58_3"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="groeb_2.html#E5:58_3"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:58"/>
<font color="Maroon" title="c8">B</font> <a href="groeb_2.html#R3" title="GROEB_2:pred.3">is_Standard_Representation_of</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:58"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="groeb_2.html#D7" target="_self" title="GROEB_2:def.7">Def7</a></i>;<br/>


</div>
