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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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#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>let </b><font color="Maroon" title="c4">P</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>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</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="E1:31"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">q</font> =  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p1'</font> = <font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c9">p2'</font> = <font color="Maroon" title="c6">p2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p1'</font> = <font color="Maroon" title="c8">p1'</font>, <font color="Maroon" title="c11">p2'</font> = <font color="Maroon" title="c9">p2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>set </b><font color="Maroon" title="c12">q1</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">q2</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">q1'</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>, <font color="Maroon" title="c15">q2'</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">q1'</font> = <font color="Maroon" title="c14">q1'</font>, <font color="Maroon" title="c17">q2'</font> = <font color="Maroon" title="c15">q2'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>
<i><font color="Green" title="E20">A2</font></i>:  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p1</font></span>)</span> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c6">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22" title="POLYRED:th.22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p1</font></span>)</span> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22" title="POLYRED:th.22">POLYRED:22</a></i>
;<br/>
<a NAME="E7:31"/>
( <font color="Maroon" title="c10">p1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  &amp; <font color="Maroon" title="c11">p2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:31"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="groeb_2.html#E27"><i><font color="Green" title="E16">Lm2</font></i></a></i>;<br/>
<a NAME="E8:31"/><b>then </b>
( <font color="Maroon" title="c16">q1'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c10">p1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  &amp; <font color="Maroon" title="c17">q2'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">p2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">q1'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c10">p1'</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">q2'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">p2'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T16" title="IDEAL_1:th.16">IDEAL_1:16</a></i>;<br/>
<a NAME="E10:31"/>
( <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q1'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c10">p1'</font> &amp; <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" title="GROEB_2:func.2">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="groeb_2.html#K1" title="GROEB_2:func.1">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q2'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">p2'</font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><a NAME="E11:31"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:31"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="groeb_2.html#E9:31"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="groeb_2.html#E28"><i><font color="Green" title="E17">Lm3</font></i></a></i>;<br/>


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