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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  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span>  st <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> holds <br/>for <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b5">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b4">g1</font>,<font color="Olive" title="b1">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b5">g2</font>,<font color="Olive" title="b1">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b1">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b4">g1</font>,<font color="Olive" title="b5">g2</font>,<font color="Olive" title="b1">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</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  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span>  st <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> holds <br/>for <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b3">g1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b4">g2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b3">g1</font>,<font color="Olive" title="b4">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</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  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <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> holds <br/>for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b3">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">g1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b3">g2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b2">g1</font>,<font color="Olive" title="b3">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><b>let </b><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">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">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> implies for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">g1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/>







<b>assume </b><a NAME="E1:69"/>
<font color="Maroon" title="c4">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>
 ; <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">g1</font>, <font color="Olive" title="b2">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">g1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/>

<a NAME="E2:69"/><b>then </b>
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font>, <font color="Olive" title="b3">h</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b3">h</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="ref" href="groeb_2.html#T28" title="GROEB_2:th.28">GROEB_2:28</a></i>;<br/>
<b>hence </b><a NAME="E3:69"/>
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <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">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">g1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="groeb_2.html#T29" title="GROEB_2:th.29">GROEB_2:29</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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