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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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#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">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span><br/> for <font color="Olive" title="b4">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span>  st <font color="Olive" title="b4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">I</font> &amp;  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b3">I</font>,<font color="Olive" title="b1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b4">G</font>,<font color="Olive" title="b1">T</font></span>)</span> holds <br/><font color="Olive" title="b4">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b3">I</font>,<font color="Olive" title="b1">T</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#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">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span><br/> for <font color="Olive" title="b3">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span>  st <font color="Olive" title="b3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">I</font> &amp;  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b2">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b3">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> holds <br/><font color="Olive" title="b3">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b2">I</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <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">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span><br/> for <font color="Olive" title="b2">G</font> being 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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> &amp;  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> holds <br/><font color="Olive" title="b2">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c4">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being 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#K29" title="POLYNOM1:func.29">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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> holds <br/><font color="Olive" title="b1">G</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c5">P</font> be  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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font> &amp;  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> implies <font color="Maroon" title="c5">P</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> )</span><br/>









<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span>)</span> or <font color="Maroon" title="c5">P</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<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> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T4" target="_self" title="GROEB_1:th.4">Th4</a></i>;<br/>
<b>assume </b><a NAME="E3:41"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">P</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font></span><br/>

<b>set </b><font color="Maroon" title="c6">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>;<br/>
<a NAME="E4:41"/><i><font color="Green" title="E35">A4</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Olive" title="b1">f</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> holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</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"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c7">f</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> implies <font color="Maroon" title="c7">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E36">A5</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c7">f</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"> <font color="Maroon" title="c7">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span><br/>

<a NAME="E2:41_1"/><b>then </b>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) </span>}</span> 
 
;<br/>
<a NAME="E3:41_1"/><b>then </b>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K3" title="GROEB_1:func.3">multiples</a> <span class="p1">(<span class="default"><a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:41"><i><font color="Green" title="E34">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">b</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font> &amp;  ex <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b1">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c8">b</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">b'</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E7:41_1"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="groeb_1.html#K2" title="GROEB_1:func.2">HT</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:41_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">p</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E9:41_1"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c9">b'</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:41_1"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E11:41_1"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c9">b'</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <font color="Maroon" title="c11">s</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:41_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="termord.html#T1" title="TERMORD:th.1">TERMORD:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c7">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E12:41_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c7">f</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> 
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:41_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/>
<a NAME="E13:41_1"/><b>then </b>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/>
<a NAME="E14:41_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c7">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">p</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:41_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="groeb_1.html#E9:41_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="groeb_1.html#E11:41_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/>
<a NAME="E15:41_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c7">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></i>;<br/>
<a NAME="E16:41_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c7">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E9:41_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/>
<b>hence </b><a NAME="E17:41_1"/>
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</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="E5:41"/><i><font color="Green" title="E36">A10</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">f</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</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"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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="E1:41_2"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</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> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2_1"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> or <font color="Maroon" title="c7">f</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="41_2_1_1"><b>suppose </b></a><a NAME="E1:41_2_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <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> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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>hence </b><a NAME="E2:41_2_1_1"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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="41_2_1_2"><b>suppose </b></a><a NAME="E1:41_2_1_2"/><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c7">f</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> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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>assume </b><a NAME="E2:41_2_1_2"/><i><font color="Green" title="E39">A13</font></i>: 
not  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</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"> contradiction</span><br/><a NAME="E3:41_2_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:41"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="groeb_1.html#E1:41_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="groeb_1.html#E1:41_2_1_2"><i><font color="Green" title="E38">A12</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">v</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E5:41_2_1_2"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></i>;<br/><a NAME="E6:41_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">v</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="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:41_2_1_2"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><a NAME="E7:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E8:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D14" title="REWRITE1:def.14">REWRITE1:def 14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:41_2_1_2"/><i><font color="Green" title="E41">A15</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c7">f</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D11" title="REWRITE1:def.11">REWRITE1:def 11</a></i>;<br/><a NAME="E11:41_2_1_2"/><i><font color="Green" title="E42">A16</font></i>: 
( <font color="Maroon" title="c9">g</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="c5">P</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:41_2_1_2"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/><a NAME="E12:41_2_1_2"/><i><font color="Green" title="E43">A17</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:41_2_1_2"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="groeb_1.html#E10:41_2_1_2"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E11:41_2_1_2"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="groeb_1.html#E20"><i><font color="Green" title="E16">Lm4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">ff</font> = <font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c12">gg</font> = <font color="Maroon" title="c10">g'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">ff</font> = <font color="Maroon" title="c11">ff</font>, <font color="Maroon" title="c14">gg</font> = <font color="Maroon" title="c12">gg</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><a NAME="E16:41_2_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c10">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">gg</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><a NAME="E17:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:41"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="groeb_1.html#E11:41_2_1_2"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="polyred.html#T59" title="POLYRED:th.59">POLYRED:59</a>, <a class="ref" href="rewrite1.html#T23" title="REWRITE1:th.23">REWRITE1:23</a></i>;<br/><a NAME="E18:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T44" title="IDEAL_1:th.44">IDEAL_1:44</a></i>;<br/><a NAME="E19:41_2_1_2"/><b>then </b><i><font color="Green" title="E44">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:41_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="ideal_1.html#T16" title="IDEAL_1:th.16">IDEAL_1:16</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">ff</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">ff</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c13">ff</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ff</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c13">ff</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font>
<b>by </b><i><a class="ref" href="algstr_1.html#D5" title="ALGSTR_1:def.5">ALGSTR_1:def 5</a></i>
;<br/><a NAME="E21:41_2_1_2"/><b>then </b>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c14">gg</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E19:41_2_1_2"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="ideal_1.html#T14" title="IDEAL_1:th.14">IDEAL_1:14</a></i>;<br/><a NAME="E22:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></i>;<br/><a NAME="E23:41_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">g'</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:41"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="groeb_1.html#E12:41_2_1_2"><i><font color="Green" title="E43">A17</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">u</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E25:41_2_1_2"/><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c10">g'</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></i>;<br/><a NAME="E26:41_2_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g'</font>,<font color="Maroon" title="c15">u</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="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E25:41_2_1_2"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></i>;<br/><b>hence </b><a NAME="E27:41_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E11:41_2_1_2"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="rewrite1.html#D5" title="REWRITE1:def.5">REWRITE1:def 5</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>
<a NAME="E6:41"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:41"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="groeb_1.html#E34"><i><font color="Green" title="E25">Lm7</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/><b>assume </b><a NAME="E1:41_3"/><i><font color="Green" title="E38">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span><br/><b>then consider </b><font color="Maroon" title="c10">a'</font>, <font color="Maroon" title="c11">b'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:41_3"/><i><font color="Green" title="E39">A22</font></i>: 
( <font color="Maroon" title="c10">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a'</font>,<font color="Maroon" title="c11">b'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><i><font color="Green" title="E40">A23</font></i>: <font color="Maroon" title="c11">b'</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:41_3"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><b>consider </b><font color="Maroon" title="c12">aa</font>, <font color="Maroon" title="c13">c'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green" title="E41">A24</font></i>: 
( <font color="Maroon" title="c12">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c13">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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="c12">aa</font>,<font color="Maroon" title="c13">c'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:41_3"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><i><font color="Green" title="E42">A25</font></i>: <font color="Maroon" title="c13">c'</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="groeb_1.html#E6:41_3"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c9">c</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c14">b'</font> = <font color="Maroon" title="c11">b'</font>, <font color="Maroon" title="c15">c'</font> = <font color="Maroon" title="c13">c'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E3:41_3"><i><font color="Green" title="E39">A22</font></i></a>, <a class="txt" href="groeb_1.html#E6:41_3"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">bb</font> = <font color="Maroon" title="c14">b'</font>, <font color="Maroon" title="c17">cc</font> = <font color="Maroon" title="c15">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">bb</font> = <font color="Maroon" title="c16">bb</font>, <font color="Maroon" title="c19">cc</font> = <font color="Maroon" title="c17">cc</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><a NAME="E11:41_3"/><i><font color="Green" title="E43">A26</font></i>: 
<font color="Maroon" title="c14">b'</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c15">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">bb</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">cc</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><a NAME="E12:41_3"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</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="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:41_3"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="rewrite1.html#T30" title="REWRITE1:th.30">REWRITE1:30</a></i>;<br/><a NAME="E13:41_3"/><b>then </b><i><font color="Green" title="E44">X</font></i>: 
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">a</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="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T32" title="REWRITE1:th.32">REWRITE1:32</a></i>;<br/><a NAME="E14:41_3"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">c</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="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:41_3"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="rewrite1.html#T30" title="REWRITE1:th.30">REWRITE1:30</a></i>;<br/><a NAME="E15:41_3"/><b>then </b>
<font color="Maroon" title="c18">bb</font>,<font color="Maroon" title="c19">cc</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:41"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="groeb_1.html#E4:41_3"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="groeb_1.html#E7:41_3"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="polyred.html#T57" title="POLYRED:th.57">POLYRED:57</a>, <a class="ref" href="rewrite1.html#T31" title="REWRITE1:th.31">REWRITE1:31</a>, <a class="txt" href="groeb_1.html#E13:41_3"><i><font color="Green" title="E44">X</font></i></a></i>;<br/><a NAME="E16:41_3"/><b>then </b>
<font color="Maroon" title="c18">bb</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="ref" href="polyred.html#D14" title="POLYRED:def.14">POLYRED:def 14</a></i>;<br/><b>hence </b><a NAME="E17:41_3"/>
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:41"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="groeb_1.html#E4:41_3"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="groeb_1.html#E7:41_3"><i><font color="Green" title="E42">A25</font></i></a>, <a class="txt" href="groeb_1.html#E11:41_3"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="polyred.html#T50" title="POLYRED:th.50">POLYRED:50</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="E8:41"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
<font color="Maroon" title="c5">P</font> <a href="groeb_1.html#R2" title="GROEB_1:pred.2">is_Groebner_basis_of</a> <font color="Maroon" title="c4">I</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:41"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</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>
