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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</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">G</font> be   <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/>







<b>assume </b><a NAME="E1:69"/>
<font color="Maroon">G</font> <a href="groeb_1.html#R1">is_Groebner_basis_wrt</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:69"/><b>then </b>
for <font color="Olive">g1</font>, <font color="Olive">g2</font>, <font color="Olive">h</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">h</font> <a href="rewrite1.html#R4">is_a_normal_form_of</a>  <a href="groeb_2.html#K3">S-Poly</a> <font color="Olive">g1</font>,<font color="Olive">g2</font>,<font color="Maroon">T</font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">h</font> <a href="funct_2.html#R2">=</a>  <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_2.html#T28">GROEB_2:28</a></i>;<br/>
<b>hence </b><a NAME="E3:69"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; not  <a href="termord.html#K3">HT</a> <font color="Olive">g1</font>,<font color="Maroon">T</font>, <a href="termord.html#K3">HT</a> <font color="Olive">g2</font>,<font color="Maroon">T</font> <a href="groeb_2.html#R1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">G</font>,<font color="Maroon">T</font> <a href="rewrite1.html#R1">reduces</a>  <a href="groeb_2.html#K3">S-Poly</a> <font color="Olive">g1</font>,<font color="Olive">g2</font>,<font color="Maroon">T</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_2.html#T29">GROEB_2:29</a></i>;<br/>


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