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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#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  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">g</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>assume </b><a NAME="E1:2"/>
<font color="Maroon" title="c4">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="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="E3:2"/><i><font color="Green" title="E2">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="c6">g</font>,<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" title="POLYRED:def.6">POLYRED:def 6</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="E5:2"/><i><font color="Green" title="E3">A2</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</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="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</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#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">/</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="groeb_1.html#E3:2"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></i>;<br/>
<a NAME="E6:2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">/</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">/</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>,<font color="Maroon" title="c8">s</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="polyred.html#T22" title="POLYRED:th.22">POLYRED:22</a></i>;<br/>
<b>hence </b><a NAME="E7:2"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="polynom1.html#K25" title="POLYNOM1:func.25">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="txt" href="groeb_1.html#E5:2"><i><font color="Green" title="E3">A2</font></i></a></i>;<br/>


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