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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</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="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/>







<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">P</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font>
 ;<br/><a NAME="E2:63_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#D3">GROEB_1:def 3</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V7">confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T12">GROEB_1:12</a></i>;<br/><a NAME="E4:63_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V4">with_UN_property</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T13">GROEB_1:13</a></i>;<br/><a NAME="E5:63_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V8">with_Church-Rosser_property</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T14">GROEB_1:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:63_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><a NAME="E2:63_1_1"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">f</font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="groeb_1.html#T15">GROEB_1:15</a></i>;<br/><b>hence </b><a NAME="E3:63_1_1"/>
<font color="Maroon">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:63"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:63_1"/>
for <font color="Olive">f</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">f</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:63_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><a NAME="E2:63_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:63_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:63_2_1"/>
<font color="Maroon">f</font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:63_2"/><b>then </b>
for <font color="Olive">b</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font> holds <br/> ex <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <font color="Olive">b'</font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> &amp; <font color="Olive">b'</font> <a href="polynom1.html#R3">divides</a> <font color="Olive">b</font> )
 <b>by </b><i><a class="ref" href="groeb_1.html#T18">GROEB_1:18</a></i>;<br/><a NAME="E4:63_2"/><b>then </b>
 <a href="groeb_1.html#K2">HT</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="groeb_1.html#K3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2">HT</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_1.html#T19">GROEB_1:19</a></i>;<br/><a NAME="E5:63_2"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> is <a href="rewrite1.html#V9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T20">GROEB_1:20</a></i>;<br/><b>hence </b><a NAME="E6:63_2"/>
<font color="Maroon">P</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D3">GROEB_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
( <font color="Maroon">P</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font> iff for <font color="Olive">f</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">f</font> <a href="groeb_2.html#R6" target="_self">has_a_Standard_Representation_of</a> <font color="Maroon">P</font>,<font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>


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