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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">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <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="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m2</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="c2">L</font>;<br/>





<b>consider </b><font color="Maroon" title="c5">T</font> being  <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/>
<a NAME="E2:7"/>
( <font color="Maroon" title="c3">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c4">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K26" title="POLYNOM1:func.26">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> )
 
<b>by </b><i><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green" title="E8">A1</font></i>: 
(  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c3">m1</font>,<font color="Maroon" title="c5">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="c2">L</font> &amp;  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">m2</font>,<font color="Maroon" title="c5">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="c2">L</font> )
 
<b>by </b><i><a class="ref" href="termord.html#T17" title="TERMORD:th.17">TERMORD:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c7">b</font> =  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m2</font> as  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">non-zero</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="termord.html#T23" title="TERMORD:th.23">TERMORD:23</a></i>;<br/>
<a NAME="E5:7"/>
(  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c3">m1</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="c2">L</font> &amp;  <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <font color="Maroon" title="c4">m2</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="c2">L</font> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E3:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="termord.html#T23" title="TERMORD:th.23">TERMORD:23</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</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="c2">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c6">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">b</font> as  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">non-zero</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></i>;<br/>
<a NAME="E8:7"/>
( <font color="Maroon" title="c3">m1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m1</font></span>)</span> &amp; <font color="Maroon" title="c4">m2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c7">b</font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom7.html#T11" title="POLYNOM7:th.11">POLYNOM7:11</a></i>;<br/>
<a NAME="E9:7"/><b>then </b>
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <font color="Maroon" title="c8">c</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="termord.html#T3" title="TERMORD:th.3">TERMORD:3</a></i>;<br/>
<b>hence </b><a NAME="E10:7"/>
 <a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c4">m2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom7.html#T10" title="POLYNOM7:th.10">POLYNOM7:10</a></i>;<br/>


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