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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being <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/> for <font color="Olive" title="b2">L</font> being 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/> for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b1">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b4">p2</font>,<font color="Olive" title="b1">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<font color="Olive" title="b1">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="b3">p1</font>,<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b1">T</font>, <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being 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/> for <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">p1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b3">p2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="b2">p1</font>,<font color="Olive" title="b3">p2</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="Olive" title="b1">L</font></span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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">p1</font>,<font color="Olive" title="b2">p2</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></span><br/><b>let </b><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a>  implies  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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> )</span><br/>







<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E57">A1</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="groeb_2.html#R1" title="GROEB_2:pred.1">are_disjoint</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/>

<a NAME="E2:68"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p2</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#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T54" target="_self" title="GROEB_3:th.54">Th54</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1"/>
( <font color="Maroon" title="c4">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> or <font color="Maroon" title="c4">p1</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><a NAME="E2:68_1_1_1"/><b>then </b><i><font color="Green" title="E60">A4</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>;<br/><a NAME="E3:68_1_1_1"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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_3.html#E1:68_1_1_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="groeb_3.html#T11" target="_self" title="GROEB_3:th.11">Th11</a></i>;<br/><b>then </b> <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<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> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="groeb_3.html#E2:68"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="groeb_3.html#E2:68_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>
<br/>.= 
 <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="polyred.html#T4" title="POLYRED:th.4">POLYRED:4</a></i>
;<br/><b>hence </b><a NAME="E5:68_1_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2"/>
<font color="Maroon" title="c4">p1</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">p1a</font> = <font color="Maroon" title="c4">p1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u</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"><font color="Maroon" title="c4">p1</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> implies <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:68_1_1_2_1"/>
<font color="Maroon" title="c7">u</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"><font color="Maroon" title="c4">p1</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span></span><br/><a NAME="E2:68_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:68_1_1_2_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:68_1_1_2"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:68_1_1_2"/><b>then </b><i><font color="Green" title="E60">A6</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"><font color="Maroon" title="c4">p1</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_1.html#T4" title="GROEB_1:th.4">GROEB_1:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u</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"><font color="Maroon" title="c5">p2</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> implies <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:68_1_1_2_2"/>
<font color="Maroon" title="c7">u</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"><font color="Maroon" title="c5">p2</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span></span><br/><a NAME="E2:68_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:68_1_1_2_2"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E7:68_1_1_2"/><b>then </b><i><font color="Green" title="E61">A7</font></i>: 
<span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c5">p2</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E8:68_1_1_2"/><b>then </b><i><font color="Green" title="E62">A8</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"><font color="Maroon" title="c5">p2</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_1.html#T4" title="GROEB_1:th.4">GROEB_1:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_3_1"/>
( <font color="Maroon" title="c5">p2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> or <font color="Maroon" title="c5">p2</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_1"/><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c5">p2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><a NAME="E2:68_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E64">A10</font></i>: 
<span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>;<br/><a NAME="E3:68_1_1_2_3_1_1"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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_3.html#E1:68_1_1_2_3_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="groeb_3.html#T11" target="_self" title="GROEB_3:th.11">Th11</a></i>;<br/><b>then </b> <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<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> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="groeb_3.html#E2:68"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="groeb_3.html#E2:68_1_1_2_3_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>
<br/>.= 
 <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="polyred.html#T4" title="POLYRED:th.4">POLYRED:4</a></i>
;<br/><b>hence </b><a NAME="E5:68_1_1_2_3_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2"/>
<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">p2a</font> = <font color="Maroon" title="c5">p2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_3_1_2_1_1"/>
(  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> or  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_1"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><a NAME="E2:68_1_1_2_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E63">A11</font></i>: 
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<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> <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#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:68"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_3_1_2_1_1_1_1_1"/>
(  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> or  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_1_1_1_1"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then </b> <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<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> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="groeb_3.html#E2:68_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>
<br/>.= 
 <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="polyred.html#T4" title="POLYRED:th.4">POLYRED:4</a></i>
;<br/><b>hence </b><a NAME="E3:68_1_1_2_3_1_2_1_1_1_1_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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="ref" href="rewrite1.html#T13" title="REWRITE1:th.13">REWRITE1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_1_1_1_2"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">rp2</font> =  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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="E3:68_1_1_2_3_1_2_1_1_1_1_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="c6">p1a</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rp2</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">p1a</font></span>)</span>, <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<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>
 <b>by </b><i><a class="ref" href="groeb_3.html#T49" target="_self" title="GROEB_3:th.49">Th49</a>, <a class="ref" href="groeb_3.html#T51" target="_self" title="GROEB_3:th.51">Th51</a></i>;<br/><a NAME="E4:68_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
 <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="c6">p1a</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">rp2</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c6">p1a</font></span>)</span>, <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="groeb_3.html#T13" target="_self" title="GROEB_3:th.13">Th13</a></i>;<br/><a NAME="E5:68_1_1_2_3_1_2_1_1_1_1_1_2"/><b>then </b>
 <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="c4">p1</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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_3.html#E2:68_1_1_2_3_1_2_1_1_1"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="groeb_3.html#T14" target="_self" title="GROEB_3:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E6:68_1_1_2_3_1_2_1_1_1_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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_3.html#E4:68_1_1_2"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="groeb_1.html#T4" title="GROEB_1:th.4">GROEB_1:4</a>, <a class="ref" href="rewrite1.html#T23" title="REWRITE1:th.23">REWRITE1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:68_1_1_2_3_1_2_1_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_2"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c8">rp1</font> =  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_3_1_2_1_1_2_1_1"/>
(  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> or  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_2_1_1_1"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><a NAME="E2:68_1_1_2_3_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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="polynom1.html#T87" title="POLYNOM1:th.87">POLYNOM1:87</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A12</font></i>:  <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p2</font></span>)</span> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:68"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="groeb_3.html#T54" target="_self" title="GROEB_3:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p2</font>
<b>by </b><i><a class="ref" href="polyred.html#T4" title="POLYRED:th.4">POLYRED:4</a></i>
;<br/><a NAME="E4:68_1_1_2_3_1_2_1_1_2_1_1_1"/>
 <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="c7">p2a</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">rp1</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c7">p2a</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="ref" href="groeb_3.html#T51" target="_self" title="GROEB_3:th.51">Th51</a></i>;<br/><b>hence </b><a NAME="E5:68_1_1_2_3_1_2_1_1_2_1_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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_3.html#E7:68_1_1_2"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="groeb_3.html#E3:68_1_1_2_3_1_2_1_1_2_1_1_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="groeb_1.html#T4" title="GROEB_1:th.4">GROEB_1:4</a>, <a class="ref" href="rewrite1.html#T23" title="REWRITE1:th.23">REWRITE1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_3_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:68_1_1_2_3_1_2_1_1_2_1_1_2"/>
 <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">rp2</font> =  <a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</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="E3:68_1_1_2_3_1_2_1_1_2_1_1_2"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c7">p2a</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c8">rp1</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#K5" title="TERMORD:func.5">HM</a> <font color="Maroon" title="c6">p1a</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c9">rp2</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:68"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="groeb_3.html#T53" target="_self" title="GROEB_3:th.53">Th53</a></i>;<br/><a NAME="E4:68_1_1_2_3_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">A13</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c5">p2</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="termord.html#K6" title="TERMORD:func.6">Red</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:68"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="groeb_3.html#E5:68_1_1_2"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="groeb_3.html#T55" target="_self" title="GROEB_3:th.55">Th55</a>, <a class="ref" href="rewrite1.html#T23" title="REWRITE1:th.23">REWRITE1:23</a></i>;<br/><a NAME="E5:68_1_1_2_3_1_2_1_1_2_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c8">rp1</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c7">p2a</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_3.html#E8:68_1_1_2"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="groeb_3.html#T51" target="_self" title="GROEB_3:th.51">Th51</a>, <a class="ref" href="rewrite1.html#T23" title="REWRITE1:th.23">REWRITE1:23</a></i>;<br/><b>hence </b><a NAME="E6:68_1_1_2_3_1_2_1_1_2_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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_3.html#E4:68_1_1_2_3_1_2_1_1_2_1_1_2"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="rewrite1.html#T17" title="REWRITE1:th.17">REWRITE1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:68_1_1_2_3_1_2_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:68_1_1_2_3_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:68_1_1_2"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:68"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <span class="p1"><a href="groeb_3.html#K1" title="GROEB_3:func.1">{</a><span class="default"><font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="groeb_3.html#K1" title="GROEB_3:func.1">}</a></span>,<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="c4">p1</font>,<font color="Maroon" title="c5">p2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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