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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  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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/>





<b>set </b><font color="Maroon" title="c4">I</font> = <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#K25" title="POLYNOM1:func.25">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>;<br/>
<b>set </b><font color="Maroon" title="c5">G</font> = <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#K25" title="POLYNOM1:func.25">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>;<br/>
<b>set </b><font color="Maroon" title="c6">R</font> =  <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E33">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c10">p</font>, <font color="Maroon" title="c11">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E34">A2</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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 href="subset_1.html#K6" title="SUBSET_1:func.6">\</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#K25" title="POLYNOM1:func.25">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> &amp; <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c11">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</font> as   <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="txt" href="groeb_1.html#E3:42_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:42_1"/>
not <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K25" title="POLYNOM1:func.25">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#E3:42_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:42_1"/><b>then </b>
<font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c10">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="txt" href="groeb_1.html#E3:42_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:42_1"/>
<font color="Maroon" title="c13">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c12">q</font>,<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#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:42_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="groeb_1.html#E3:42_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">g</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><b> such that </b><br/><a NAME="E10:42_1"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K25" title="POLYNOM1:func.25">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> &amp; <font color="Maroon" title="c13">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/><a NAME="E11:42_1"/>
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:42_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:42_1"/><b>then </b>
<font color="Maroon" title="c13">p</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:42_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></i>;<br/><b>hence </b><a NAME="E13:42_1"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:42"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
 <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<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="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></i>;<br/>
<a NAME="E3:42"/>
 <a href="polynom1.html#K25" title="POLYNOM1:func.25">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E4:42"/><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#K25" title="POLYNOM1:func.25">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#K29" title="POLYNOM1:func.29">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="E5:42"/><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#K25" title="POLYNOM1:func.25">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> <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#K25" title="POLYNOM1:func.25">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="ref" href="ideal_1.html#T44" title="IDEAL_1:th.44">IDEAL_1:44</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
<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#K25" title="POLYNOM1:func.25">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> <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"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">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>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:42"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></i>;<br/>


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