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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/><b>let </b><font color="Maroon" title="c6">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>;<br/>









<b>assume </b><a NAME="E1:71"/>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green" title="E55">A2</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c5">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="c7">b</font> &amp; <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E3:71"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/>
<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E5:71"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E56">A3</font></i>:  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> = 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">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="p5">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">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="p5">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T1" target="_self" title="POLYRED:th.1">Th1</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p4">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></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="c5">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="p0">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p3">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p4">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></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="c5">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="p0">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T1" target="_self" title="POLYRED:th.1">Th1</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></i>
;<br/>
<a NAME="E8:71"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="polyred.html#E3:71"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/>
<a NAME="E9:71"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font> <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>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E10:71"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<b>by </b><i><a class="txt" href="polyred.html#E3:71"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/>
<a NAME="E11:71"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font> <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>
 
<b>by </b><i><a class="ref" href="termord.html#T17" title="TERMORD:th.17">TERMORD:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_3"/>
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <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>
 ;<br/><b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font> = 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="c5">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>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="polyred.html#E11:71"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></i>
;<br/><b>hence </b><a NAME="E3:71_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E13:71"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="c5">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> <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>
 
<b>by </b><i><a class="txt" href="polyred.html#E9:71"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E14:71"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></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="c5">p</font>,<font color="Maroon" title="c2">T</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>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></i>;<br/>
<a NAME="E15:71"/><b>then </b><i><font color="Green" title="E60">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">b</font></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="c5">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">non-zero</a>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></i>;<br/>
<a NAME="E16:71"/><i><font color="Green" title="E61">A8</font></i>: 
<font color="Maroon" title="c5">p</font> is <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>
 
<b>by </b><i><a class="txt" href="polyred.html#E10:71"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/>
<font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E7:71"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T83" title="POLYNOM1:th.83">POLYNOM1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">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="c5">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="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></i>
;<br/>
<b>then </b> <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> = 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="polyred.html#E15:71"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="polyred.html#T21" target="_self" title="POLYRED:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="txt" href="polyred.html#E5:71"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="polyred.html#E16:71"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="polyred.html#T15" target="_self" title="POLYRED:th.15">Th15</a></i>
;<br/>
<b>hence </b><a NAME="E19:71"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="polyred.html#E3:71"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></i>;<br/>


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