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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>   <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b2">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b3">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#K11" title="POLYNOM1:func.11">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#K11" title="POLYNOM1:func.11">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="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/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>   <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">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#K11" title="POLYNOM1:func.11">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#K11" title="POLYNOM1:func.11">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="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/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K11" title="POLYNOM1:func.11">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#K11" title="POLYNOM1:func.11">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="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/><span class="kw">let </span><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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b2">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b2">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/><span class="kw">let </span><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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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/>









<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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/>

<span class="kw">assume </span><a NAME="E2:35"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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/>

<span class="kw">thus </span><a NAME="E3:35"/>
 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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">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#NM2" title="POLYNOM1:NM.2">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/>
<span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <font color="Maroon" title="c7">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/>

<span class="kw">assume </span><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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/>

<span class="kw">assume </span><a NAME="E2:35_1"/>
 not <font color="Maroon" title="c7">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c7">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#NM2" title="POLYNOM1:NM.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM2" title="POLYNOM1:NM.2">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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:35_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#NM2" title="POLYNOM1:NM.2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span></span><br/><a NAME="E2:35_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g9</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g9</font> &amp; <font color="Olive" title="b1">g9</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">g9</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/><span class="kw">hence </span><a NAME="E3:35_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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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#NM2" title="POLYNOM1:NM.2">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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E7:35_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:35_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T31" title="POLYRED:th.31">POLYRED:31</a></span>;<br/>
<a NAME="E9:35_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 ex <font color="Olive" title="b1">p9</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <br/>( <font color="Olive" title="b1">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Olive" title="b1">p9</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">p9</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E11:35_1"/>
<font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E13:35_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">u</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E15:35_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E16:35_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/>
<a NAME="E17:35_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <font color="Maroon" title="c10">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c12">u</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></span>;<br/>
<a NAME="E18:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">u</font> = <font color="Maroon" title="c12">u</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E21:35_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E38">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_2"/>
<font color="Maroon" title="c14">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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c11">q</font>,<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></span>;<br/><a NAME="E3:35_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E15:35_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">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><span class="kw"> such that </span><br/><a NAME="E24:35_1"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="groeb_1.html#T1" target="_self" title="GROEB_1:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">uu</font> = <font color="Maroon" title="c13">u</font>, <font color="Maroon" title="c17">pp</font> = <font color="Maroon" title="c10">p</font>, <font color="Maroon" title="c18">mm</font> = <font color="Maroon" title="c15">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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">uu</font> = <font color="Maroon" title="c16">uu</font>, <font color="Maroon" title="c20">pp</font> = <font color="Maroon" title="c17">pp</font>, <font color="Maroon" title="c21">mm</font> = <font color="Maroon" title="c18">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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> ;<br/>
<a NAME="E27:35_1"/>
<font color="Maroon" title="c21">mm</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">uu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E15:35_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></span>;<br/>
<a NAME="E28:35_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">mm</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">uu</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T13" title="IDEAL_1:th.13">IDEAL_1:13</a></span>;<br/>
<a NAME="E29:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c20">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="c21">mm</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">uu</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></span>;<br/>
<a NAME="E30:35_1"/>
<font color="Maroon" title="c21">mm</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">uu</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/>
<a NAME="E31:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">pp</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">mm</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c19">uu</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="groeb_1.html#E4"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E32:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E29:35_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>;<br/>
<a NAME="E33:35_1"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="polyred.html#T43" title="POLYRED:th.43">POLYRED:43</a></span>;<br/>
<a NAME="E34:35_1"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/>
<a NAME="E35:35_1"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c10">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/>
<a NAME="E36:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A19</font></span>: 
 <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>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c10">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/>
<a NAME="E37:35_1"/>
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c10">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:35_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/>
<a NAME="E38:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c10">p</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/>
<a NAME="E39:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c10">p</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:35_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></span>;<br/>
<a NAME="E40:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 <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>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:35_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E21:35_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="polyred.html#T40" title="POLYRED:th.40">POLYRED:40</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_1_3_1"/>
( <font color="Maroon" title="c11">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) or <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="35_1_3_1_1"><span class="kw">case </span></a><a NAME="E1:35_1_3_1_1"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c11">q</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><a NAME="E3:35_1_3_1_1"/>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c22">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_3_1_1"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E4:35_1_3_1_1"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c22">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><a NAME="E5:35_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c22">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="c10">p</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:35_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="polyred.html#T42" title="POLYRED:th.42">POLYRED:42</a></span>;<br/><a NAME="E6:35_1_3_1_1"/>
 <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>) <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="c22">q</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:35_1"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><a NAME="E7:35_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A23</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c22">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="c10">p</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_3_1_1"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:35_1_3_1_1_1"/>
 not <font color="Maroon" title="c22">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:35_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:35_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a></span>;<br/><a NAME="E3:35_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:35_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:35_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="polyred.html#T29" title="POLYRED:th.29">POLYRED:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">u9</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E10:35_1_3_1_1"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c23">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E11:35_1_3_1_1"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c23">u9</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">q9</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E13:35_1_3_1_1"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c24">q9</font>,<font color="Maroon" title="c23">u9</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:35_1_3_1_1"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></span>;<br/><a NAME="E14:35_1_3_1_1"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><a NAME="E15:35_1_3_1_1"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E16:35_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A28</font></span>: 
 <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>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><a NAME="E17:35_1_3_1_1"/><span class="lab"><font color="Green" title="E54">A29</font></span>: 
<font color="Maroon" title="c22">q</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c24">q9</font>,<font color="Maroon" title="c23">u9</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_3_1_1"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">s</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E19:35_1_3_1_1"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
<font color="Maroon" title="c25">s</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c23">u9</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="c22">q</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">and </span><br/><a NAME="E20:35_1_3_1_1"/>
<font color="Maroon" title="c24">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c22">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c22">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="c23">u9</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="c25">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c23">u9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">qq</font> = <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c23">u9</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="c25">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c23">u9</font></span>)</span></span>)</span>;<br/><a NAME="E21:35_1_3_1_1"/>
<font color="Maroon" title="c23">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:35_1_3_1_1"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/><a NAME="E22:35_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c23">u9</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="c25">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c23">u9</font></span>)</span></span>)</span>,<font color="Maroon" title="c23">u9</font>, <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>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_1_3_1_1"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="txt" href="#E19:35_1_3_1_1"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E14:35_1_3_1_1"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="txt" href="#E16:35_1_3_1_1"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/><a NAME="E23:35_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c23">u9</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="c25">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c23">u9</font></span>)</span></span>)</span>,<font color="Maroon" title="c23">u9</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></span>;<br/><a NAME="E24:35_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c23">u9</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E25:35_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E10:35_1_3_1_1"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_3_1_2"><span class="kw">case </span></a><a NAME="E1:35_1_3_1_2"/>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A31</font></span>: <font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:35_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c10">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c13">u</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T21" title="POLYNOM1:th.21">POLYNOM1:21</a></span>
<br/>.= 
<font color="Maroon" title="c10">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></span>
<br/>.= 
<font color="Maroon" title="c10">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T23" title="POLYNOM1:th.23">POLYNOM1:23</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_1_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:35_1_3_1_2_2"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:35_1_3_1_2_2"/>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E3:35_1_3_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_3_1_2"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="txt" href="#E1:35_1_3_1_2_2"><span class="lab"><font color="Green" title="E47">A32</font></span></a>, <a class="ref" href="polyred.html#T5" title="POLYRED:th.5">POLYRED:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m</font> = <font color="Maroon" title="c15">m</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">pp</font> = <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c13">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="c22">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="c13">u</font></span>)</span></span>)</span>;<br/><a NAME="E5:35_1_3_1_2"/>
 <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>) <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="c22">m</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_3_1_2"><span class="lab"><font color="Green" title="E46">A31</font></span></a>, <a class="ref" href="termord.html#T31" title="TERMORD:th.31">TERMORD:31</a></span>;<br/><a NAME="E6:35_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c13">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="c22">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="c13">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">u</font>, <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>),<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:35_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E34:35_1"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="txt" href="#E36:35_1"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/><a NAME="E7:35_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R8" title="POLYRED:pred.8">top_reduces_to</a> <font color="Maroon" title="c10">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><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>)</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="c13">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="c22">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="c13">u</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D10" title="POLYRED:def.10">POLYRED:def 10</a></span>;<br/><a NAME="E8:35_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="polyred.html#R9" title="POLYRED:pred.9">is_top_reducible_wrt</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D11" title="POLYRED:def.11">POLYRED:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:35_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E15:35_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="polyred.html#D12" title="POLYRED:def.12">POLYRED:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E42:35_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
