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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</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>, <font color="Olive" title="b3">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds <br/> (  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b3">q</font> &amp;  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Olive" title="b3">q</font></span>)</span> )</span><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="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds <br/> (  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">q</font> &amp;  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Olive" title="b2">q</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">q</font></span>)</span> )</span><br/>





<b>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c6">q'</font> = <font color="Maroon" title="c4">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c5">p'</font>, <font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c6">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> ;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">p'</font>
 
<b>by </b><i><a class="txt" href="polyred.html#E12"><i><font color="Green" title="E12">Lm7</font></i></a></i>;<br/>
<a NAME="E4:16"/>
 <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">q'</font>
 
<b>by </b><i><a class="txt" href="polyred.html#E12"><i><font color="Green" title="E12">Lm7</font></i></a></i>;<br/>
<a NAME="E5:16"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
( <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">p'</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c7">p'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">q'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="polyred.html#E3:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green" title="E16">A3</font></i>: 
(  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">q'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">p'</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">q'</font> &amp;  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">q'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">q'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>;<br/>
<a NAME="E7:16"/>
<font color="Maroon" title="c7">p'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><a NAME="E8:16"/>
(  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c4">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E5:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="polyred.html#E6:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="polyred.html#E12"><i><font color="Green" title="E12">Lm7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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