<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#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/> for <font color="Olive" title="b3">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span>  st <font color="Olive" title="b3">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Olive" title="b1">T</font> holds <br/>for <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">g2</font>, <font color="Olive" title="b6">h</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b5">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b6">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="b4">g1</font>,<font color="Olive" title="b5">g2</font>,<font color="Olive" title="b1">T</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being 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/> for <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span>  st <font color="Olive" title="b2">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> holds <br/>for <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font>, <font color="Olive" title="b5">h</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b5">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="b3">g1</font>,<font color="Olive" title="b4">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b1">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> holds <br/>for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font>, <font color="Olive" title="b4">h</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b2">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b3">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b4">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="b2">g1</font>,<font color="Olive" title="b3">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><span class="kw">let </span><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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font> implies 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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/>

<span class="kw">set </span><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/>
<a NAME="E2:40"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1"/>
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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2"/>
<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c4">G</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><a NAME="E3:40_1_2"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <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> is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="groeb_1.html#D3" title="GROEB_1:def.3">GROEB_1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E23">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_2_1_1"/>
not  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1_2_1_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D5" title="REWRITE1:def.5">REWRITE1:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c9">f2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:40_1_2_1_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:40_1_2_1_1"/>
<font color="Maroon" title="c9">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_2_1_1"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c9">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2_1_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E27">A8</font></span>: <font color="Maroon" title="c8">f1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_2_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
<br/>.= 
 <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a></span>
;<br/><a NAME="E9:40_1_2_1_1"/>
not <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:40_1_2_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:40_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_2_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c7">g1</font>, <font color="Maroon" title="c8">g2</font>, <font color="Maroon" title="c9">h</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c9">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="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">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> implies <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:40_1_2_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c7">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> &amp; <font color="Maroon" title="c8">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> )
 <span class="kw">and </span><br/><a NAME="E3:40_1_2_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c9">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="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span><br/><a NAME="E4:40_1_2_1"/>
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="groeb_2.html#T22" target="_self" title="GROEB_2:th.22">Th22</a></span>;<br/><a NAME="E5:40_1_2_1"/><span class="kw">then </span>
 <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="c7">g1</font>,<font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="groeb_1.html#T15" title="GROEB_1:th.15">GROEB_1:15</a></span>;<br/><a NAME="E6:40_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A11</font></span>: 
 <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c2">T</font>, <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></span>;<br/><a NAME="E7:40_1_2_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="c7">g1</font>,<font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c9">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></span>;<br/><a NAME="E8:40_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font>, <a href="groeb_2.html#K3" title="GROEB_2:func.3">S-Poly</a> <font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">g2</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></span>;<br/><a NAME="E9:40_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c9">h</font>, <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_2_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="rewrite1.html#T31" title="REWRITE1:th.31">REWRITE1:31</a></span>;<br/><a NAME="E10:40_1_2_1"/>
<font color="Maroon" title="c9">h</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:40_1_2_1"/>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E1:40_1_2_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E9:40_1_2_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a>, <a class="ref" href="rewrite1.html#D19" title="REWRITE1:def.19">REWRITE1:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:40_1_2"/>
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#NM2" title="POLYNOM1:NM.2">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#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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