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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#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/><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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E21">A1</font></i>: 
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="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>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">R</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>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</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/><b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
(  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><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></span>)</span> or  not  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><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></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>case </b></a><a NAME="E1:37_1_1_1_1"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><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></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_1"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a>  <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>
 <b>by </b><i><a class="ref" href="rewrite1.html#D14" title="REWRITE1:def.14">REWRITE1:def 14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>case </b></a><a NAME="E1:37_1_1_1_2"/>
not  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><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></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1_2"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a>  <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>
 <b>by </b><i><a class="ref" href="rewrite1.html#T47" title="REWRITE1:th.47">REWRITE1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c8">q</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="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">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>
 <b>by </b><i><a class="ref" href="rewrite1.html#D11" title="REWRITE1:def.11">REWRITE1:def 11</a></i>;<br/><a NAME="E5:37_1"/>
( <font color="Maroon" title="c8">q</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a>  <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> &amp;  <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="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c8">q</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E4:37_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">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_2.html#E24"><i><font color="Green" title="E14">Lm1</font></i></a></i>;<br/><a NAME="E7:37_1"/>
<font color="Maroon" title="c9">q</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_2.html#E1:37"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="groeb_2.html#E1:37_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="groeb_2.html#E4:37_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E8:37_1"/>
 <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="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">g2</font>,<font color="Maroon" title="c2">T</font>, <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_2.html#E4:37_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
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> 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#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ;<br/>


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