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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> ;<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>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  <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="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</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/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/><div class="add"><a NAME="E2:43_1_1"/>
 <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <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>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E3:43_1_1"/><b>then </b>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</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>
 <b>by </b><i><a class="ref" href="ideal_1.html#T7" title="IDEAL_1:th.7">IDEAL_1:7</a></i>;<br/><a NAME="E4:43_1_1"/><b>then </b>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:43_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="ideal_1.html#T44" title="IDEAL_1:th.44">IDEAL_1:44</a></i>;<br/><b>hence </b><a NAME="E5:43_1_1"/>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> ,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:43_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="groeb_1.html#T30" target="_self" title="GROEB_1:th.30">Th30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as  <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>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:43_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c5">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<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#T10" target="_self" title="GROEB_1:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E4:43_1_2"/>
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> ,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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