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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 <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  <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; ( for <font color="Olive" title="b5">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="b5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">I</font> holds <br/><font color="Olive" title="b5">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Olive" title="b4">G</font>,<font color="Olive" title="b1">T</font> ) holds <br/>for <font color="Olive" title="b5">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="b5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">I</font> holds <br/><font color="Olive" title="b5">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Olive" title="b4">G</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 <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  <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; ( for <font color="Olive" title="b4">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">I</font> holds <br/><font color="Olive" title="b4">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Olive" title="b3">G</font>,<font color="Maroon" title="c2">T</font> ) holds <br/>for <font color="Olive" title="b4">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">I</font> holds <br/><font color="Olive" title="b4">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Olive" title="b3">G</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 <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  <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; ( for <font color="Olive" title="b3">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/><font color="Olive" title="b3">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font> ) holds <br/>for <font color="Olive" title="b3">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/><font color="Olive" title="b3">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c4">I</font> be  <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  <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; ( for <font color="Olive" title="b2">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><font color="Olive" title="b2">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font> ) holds <br/>for <font color="Olive" title="b2">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> holds <br/><font color="Olive" title="b2">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c5">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <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; ( for <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><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> ) implies for <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>









<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E29">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"> (  ex <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <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; not <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> ) or for <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>assume </b><a NAME="E2:38"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><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>
 ; <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">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span><br/>

<b>thus </b><a NAME="E3:38"/>
for <font color="Olive" title="b1">f</font> being <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  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/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">f</font> be  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> implies <font color="Maroon" title="c6">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c6">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"> <font color="Maroon" title="c6">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font></span><br/>

<b>set </b><font color="Maroon" title="c7">H</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Olive" title="b1">g</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:38_1"/>
not <font color="Maroon" title="c6">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</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"> contradiction</span><br/>

<a NAME="E3:38_1"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Olive" title="b1">g</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</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#E1:38_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Olive" title="b1">g</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> ) </span>}</span>  implies <font color="Maroon" title="c8">u</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> )</span><br/><b>assume </b><a NAME="E1:38_1_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Olive" title="b1">g</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</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="c8">u</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></span><br/><b>then consider </b><font color="Maroon" title="c9">g'</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> &amp; <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> &amp; not <font color="Maroon" title="c9">g'</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> )
 ;<br/><b>thus </b><a NAME="E4:38_1_1"/>
<font color="Maroon" title="c8">u</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>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:38_1_1"><i><font color="Green" title="E33">A5</font></i></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>
<b>then reconsider </b><font color="Maroon" title="c8">H</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> : ( <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Olive" title="b1">g</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font> ) </span>}</span>  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#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="txt" href="groeb_1.html#E3:38_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E7:38_1"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H</font> &amp; ( for <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H</font> holds <br/><font color="Maroon" title="c9">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">T</font> ) )
 <b>by </b><i><a class="ref" href="polyred.html#T31" title="POLYRED:th.31">POLYRED:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">p'</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E9:38_1"/><i><font color="Green" title="E34">A7</font></i>: 
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; not <font color="Maroon" title="c10">p'</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_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#E7:38_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c9">p</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="txt" href="groeb_1.html#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<a NAME="E11:38_1"/>
<font color="Maroon" title="c11">p</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#E2:38"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="groeb_1.html#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">q</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="E13:38_1"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c12">q</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/>
<b>consider </b><font color="Maroon" title="c13">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="E15:38_1"/><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c11">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:38_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">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="E17:38_1"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></i>;<br/>
<a NAME="E18:38_1"/><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c13">u</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#E17:38_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">u</font> = <font color="Maroon" title="c13">u</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E20:38_1"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c12">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="polyred.html#T43" title="POLYRED:th.43">POLYRED:43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">m</font> being   <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E22:38_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">uu</font> = <font color="Maroon" title="c15">u</font>, <font color="Maroon" title="c18">pp</font> = <font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c19">mm</font> = <font color="Maroon" title="c16">m</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="c20">uu</font> = <font color="Maroon" title="c17">uu</font>, <font color="Maroon" title="c21">pp</font> = <font color="Maroon" title="c18">pp</font>, <font color="Maroon" title="c22">mm</font> = <font color="Maroon" title="c19">mm</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="E25:38_1"/>
<font color="Maroon" title="c22">mm</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c20">uu</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#E1:38"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E26:38_1"/><b>then </b>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">mm</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c20">uu</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="ref" href="ideal_1.html#T13" title="IDEAL_1:th.13">IDEAL_1:13</a></i>;<br/>
<a NAME="E27:38_1"/><b>then </b><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c21">pp</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c22">mm</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c20">uu</font></span>)</span></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#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></i>;<br/>
<a NAME="E28:38_1"/>
<font color="Maroon" title="c22">mm</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c20">uu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E29:38_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">pp</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">mm</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c20">uu</font></span>)</span>
 
<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="E30:38_1"/><b>then </b><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c12">q</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#E22:38_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="groeb_1.html#E27:38_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>;<br/>
<a NAME="E31:38_1"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c11">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="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E32:38_1"/><b>then </b>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p</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="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/>
<a NAME="E33:38_1"/><b>then </b><i><font color="Green" title="E44">A17</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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="c11">p</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">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="E35:38_1"/><i><font color="Green" title="E45">A18</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></i>;<br/>
<a NAME="E36:38_1"/>
<font color="Maroon" title="c23">b</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="c11">p</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E35:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/>
<a NAME="E37:38_1"/><b>then </b><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c23">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c23">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="c11">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"> contradiction</span><br/><a NAME="E2:38_1_2"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E35:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></i>;<br/><a NAME="E3:38_1_2"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E4:38_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</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="E39:38_1"/><b>then </b>
<font color="Maroon" title="c23">b</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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#E37:38_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/>
<a NAME="E40:38_1"/><b>then </b><i><font color="Green" title="E47">A20</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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="c12">q</font>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E33:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="groeb_1.html#E35:38_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="polyred.html#T40" title="POLYRED:th.40">POLYRED:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_3_1"/>
( <font color="Maroon" title="c12">q</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> or <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_3_1_1"><b>case </b></a><a NAME="E1:38_1_3_1_1"/><i><font color="Green" title="E48">A21</font></i>: 
<font color="Maroon" title="c12">q</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"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c12">q</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E3:38_1_3_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c24">q</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#E1:38_1_3_1_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/><a NAME="E4:38_1_3_1_1"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</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="c24">q</font>
 <b>by </b><i><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/><a NAME="E5:38_1_3_1_1"/><i><font color="Green" title="E50">A23</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</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#E40:38_1"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/><a NAME="E6:38_1_3_1_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="groeb_1.html#E4:38_1_3_1_1"><i><font color="Green" title="E49">A22</font></i></a>, <a class="ref" href="polyred.html#T42" title="POLYRED:th.42">POLYRED:42</a></i>;<br/><a NAME="E7:38_1_3_1_1"/><b>then </b><i><font color="Green" title="E51">A24</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</font>,<font color="Maroon" title="c2">T</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="c11">p</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:38_1_3_1_1"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_3_1_1_1"/>
not <font color="Maroon" title="c24">q</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</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"> contradiction</span><br/><a NAME="E2:38_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E30:38_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/><a NAME="E3:38_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c24">q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:38_1"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:38_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:38_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="polyred.html#T29" title="POLYRED:th.29">POLYRED:29</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><b>then consider </b><font color="Maroon" title="c25">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="E10:38_1_3_1_1"/><i><font color="Green" title="E52">A25</font></i>: 
( <font color="Maroon" title="c25">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c24">q</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">q'</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="E12:38_1_3_1_1"/><i><font color="Green" title="E53">A26</font></i>: 
<font color="Maroon" title="c24">q</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c26">q'</font>,<font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:38_1_3_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></i>;<br/><a NAME="E13:38_1_3_1_1"/><i><font color="Green" title="E54">A27</font></i>: 
<font color="Maroon" title="c24">q</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c26">q'</font>,<font color="Maroon" title="c25">u'</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</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#E12:38_1_3_1_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></i>;<br/><a NAME="E14:38_1_3_1_1"/><b>then </b><i><font color="Green" title="E55">A28</font></i>: 
( <font color="Maroon" title="c24">q</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> &amp; <font color="Maroon" title="c25">u'</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> &amp;  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">q</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="c24">q</font> &amp;  ex <font color="Olive" title="b1">s</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">s</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font></span>)</span> <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="c24">q</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c26">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</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="c24">q</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="c24">q</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="c25">u'</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="Olive" title="b1">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c25">u'</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c27">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="E16:38_1_3_1_1"/><i><font color="Green" title="E56">A29</font></i>: 
( <font color="Maroon" title="c27">s</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font></span>)</span> <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="c24">q</font>,<font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c26">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</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="c24">q</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="c24">q</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="c25">u'</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="c27">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c25">u'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E13:38_1_3_1_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c28">qq</font> = <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c25">u'</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="c27">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c25">u'</font></span>)</span></span>)</span>;<br/><a NAME="E17:38_1_3_1_1"/><i><font color="Green" title="E57">A30</font></i>: 
<font color="Maroon" title="c11">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="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E18:38_1_3_1_1"/><b>then </b>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p</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="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></i>;<br/><a NAME="E19:38_1_3_1_1"/><b>then </b>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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="c11">p</font>
 <b>by </b><i><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></i>;<br/><a NAME="E20:38_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c25">u'</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="c27">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c25">u'</font></span>)</span></span>)</span>,<font color="Maroon" title="c25">u'</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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#E7:38_1_3_1_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="groeb_1.html#E14:38_1_3_1_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="groeb_1.html#E16:38_1_3_1_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="groeb_1.html#E17:38_1_3_1_1"><i><font color="Green" title="E57">A30</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/><a NAME="E21:38_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c25">u'</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="c27">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c25">u'</font></span>)</span></span>)</span>,<font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></i>;<br/><a NAME="E22:38_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c25">u'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E23:38_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="groeb_1.html#E10:38_1_3_1_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</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="38_1_3_1_2"><b>case </b></a><a NAME="E1:38_1_3_1_2"/>
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then </b><i><font color="Green" title="E48">A31</font></i>: <font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E22:38_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c11">p</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon" title="c11">p</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></i>
<br/>.= 
<font color="Maroon" title="c11">p</font>
<b>by </b><i><a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_3_1_2_2"/><i><font color="Green" title="E49">A32</font></i>: 
<font color="Maroon" title="c16">m</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"> contradiction</span><br/><a NAME="E2:38_1_3_1_2_2"/>
<font color="Maroon" title="c11">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="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><b>hence </b><a NAME="E3:38_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:38_1_3_1_2"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="groeb_1.html#E1:38_1_3_1_2_2"><i><font color="Green" title="E49">A32</font></i></a>, <a class="ref" href="polyred.html#T5" title="POLYRED:th.5">POLYRED: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><b>then reconsider </b><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c16">m</font> as  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/><a NAME="E5:38_1_3_1_2"/><i><font color="Green" title="E49">A33</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:38_1_3_1_2"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="termord.html#T31" title="TERMORD:th.31">TERMORD:31</a></i>;<br/><b>set </b><font color="Maroon" title="c25">pp</font> = <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c15">u</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"><span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span>;<br/><a NAME="E6:38_1_3_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c15">u</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"><span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">u</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c11">p</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#E18:38_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="groeb_1.html#E31:38_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="txt" href="groeb_1.html#E33:38_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="groeb_1.html#E5:38_1_3_1_2"><i><font color="Green" title="E49">A33</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/><a NAME="E7:38_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c11">p</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="c11">p</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="c11">p</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="c15">u</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"><span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c24">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c15">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></i>;<br/><a NAME="E8:38_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c15">u</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></i>;<br/><b>hence </b><a NAME="E9:38_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:38_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="groeb_1.html#E15:38_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E42:38_1"/>
contradiction
 ; <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>
