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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>   <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a>  <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">L</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b3">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font><br/> for <font color="Olive" title="b6">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>  st <font color="Olive" title="b4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b6">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b6">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">P</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b5">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st <font color="Olive" title="b3">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b5">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b3">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b5">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">P</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</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><br/> for <font color="Olive" title="b4">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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>  st <font color="Olive" title="b2">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b4">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b2">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b4">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</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><br/> for <font color="Olive" title="b3">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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>  st <font color="Olive" title="b1">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b1">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b3">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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>  st <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b1">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b1">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">m</font> be   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:81"/>
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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="E3:81"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E4:81"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">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="E6:81"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">b9</font> = <font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span>;<br/>
<a NAME="E7:81"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:81_1"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E2:81_1"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c7">m</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="E3:81_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D6" title="POLYNOM7:def.6">POLYNOM7:def 6</a></span>;<br/><span class="kw">assume </span><a NAME="E4:81_1"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T23" title="TERMORD:th.23">TERMORD:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T7" target="_self" title="POLYRED:th.7">Th7</a></span>
;<br/><a NAME="E6:81_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E3:81_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:81_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:81"/>
<font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<a NAME="E10:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A8</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:81"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<a NAME="E11:81"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="polyred.html#E74"><span class="lab"><font color="Green" title="E59">Lm18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">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="E13:81"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">and </span><br/><a NAME="E14:81"/><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c8">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:81"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/>
<a NAME="E16:81"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:81"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="pre_poly.html#T35" title="PRE_POLY:th.35">PRE_POLY:35</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E17:81"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c5">f</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#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/>
<a NAME="E19:81"/>
( <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c12">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/>
<a NAME="E20:81"/><span class="kw">then </span>
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:81"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="#E16:81"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/>
<a NAME="E21:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A14</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T23" title="TERMORD:th.23">TERMORD:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T7" target="_self" title="POLYRED:th.7">Th7</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p5">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></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="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:81"><span class="lab"><font color="Green" title="E73">A14</font></span></a>, <a class="ref" href="polyred.html#T18" target="_self" title="POLYRED:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p4">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T11" target="_self" title="POLYRED:th.11">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p5">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T22" target="_self" title="POLYRED:th.22">Th22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p4">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c7">m</font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p4">(<span class="default"><a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c7">m</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c7">m</font></span>)</span></span>)</span> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T23" title="TERMORD:th.23">TERMORD:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></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="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T11" title="POLYNOM7:th.11">POLYNOM7:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T12" target="_self" title="POLYRED:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T6" target="_self" title="POLYRED:th.6">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T85" title="POLYNOM1:th.85">POLYNOM1:85</a></span>
<br/>.= 
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E25:81"/>
<font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c7">m</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E14:81"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E21:81"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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