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<b>let </b><font color="Maroon" title="c1">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b3">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span> holds <br/> ( <font color="Olive" title="b3">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Olive" title="b1">T</font> iff for <font color="Olive" title="b4">f</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="Olive" title="b2">L</font>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b4">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b1">T</font> )</span><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#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span> holds <br/> ( <font color="Olive" title="b2">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> iff for <font color="Olive" title="b3">f</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="Olive" title="b1">L</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b3">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font> )</span><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#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> iff for <font color="Olive" title="b2">f</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 <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b2">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font> )</span><br/><b>let </b><font color="Maroon" title="c4">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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> iff for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>







<div><i><font color="Green" title="E33">A1</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" title="c4">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:63_1"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#D3" title="GROEB_1:def.3">GROEB_1:def 3</a></i>;<br/><a NAME="E3:63_1"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T12" title="GROEB_1:th.12">GROEB_1:12</a></i>;<br/><a NAME="E4:63_1"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V8" title="REWRITE1:attr.8">with_Church-Rosser_property</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T13" title="GROEB_1:th.13">GROEB_1:13</a>, <a class="ref" href="groeb_1.html#T14" title="GROEB_1:th.14">GROEB_1:14</a></i>;<br/><b>hence </b><a NAME="E5:63_1"/>
for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T43" target="_self" title="GROEB_2:th.43">Th43</a>, <a class="ref" href="groeb_1.html#T15" title="GROEB_1:th.15">GROEB_1:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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"/>
for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:63_2"/><b>then </b>
for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T44" target="_self" title="GROEB_2:th.44">Th44</a></i>;<br/><a NAME="E3:63_2"/><b>then </b>
for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>,<font color="Maroon" title="c2">T</font> holds <br/> ex <font color="Olive" title="b2">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b2">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b2">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="ref" href="groeb_1.html#T18" title="GROEB_1:th.18">GROEB_1:18</a></i>;<br/><a NAME="E4:63_2"/><b>then </b>
 <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_1.html#T19" title="GROEB_1:th.19">GROEB_1:19</a></i>;<br/><a NAME="E5:63_2"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 <b>by </b><i><a class="ref" href="groeb_1.html#T20" title="GROEB_1:th.20">GROEB_1:20</a></i>;<br/><b>hence </b><a NAME="E6:63_2"/>
<font color="Maroon" title="c4">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D3" title="GROEB_1:def.3">GROEB_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
( <font color="Maroon" title="c4">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> iff for <font color="Olive" title="b1">f</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 <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="groeb_2.html#R6" title="GROEB_2:pred.6">has_a_Standard_Representation_of</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:63"><i><font color="Green" title="E33">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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