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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">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="c5">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c3">L</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c4">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> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">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/>
<b>set </b><font color="Maroon" title="c6">c</font> = <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">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="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></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="groeb_1.html#E2:57"><i><font color="Green" title="E43">A2</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="E3:57_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">c</font> = <span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  as  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">non-zero</a> <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D14" title="RLVECT_1:def.14">RLVECT_1:def 14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font>;<br/>
<b>take </b>
<font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c9">pp</font> = <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c10">cc</font> = <font color="Maroon" title="c7">c</font> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</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="c3">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="c11">pp</font> = <font color="Maroon" title="c9">pp</font>, <font color="Maroon" title="c12">cc</font> = <font color="Maroon" title="c10">cc</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="c3">L</font></span>)</span> ;<br/>
<font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="ref" href="polynom7.html#T27" title="POLYNOM7:th.27">POLYNOM7:27</a></i>
<br/>.= 
<font color="Maroon" title="c12">cc</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c11">pp</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="E8:57"/>
<font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:57"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>

<a NAME="E9:57"/><i><font color="Green" title="E44">A3</font></i>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="polyred.html#T21" title="POLYRED:th.21">POLYRED:21</a></i>;<br/>
<b>then </b> <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c7">c</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c4">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="c4">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="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</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="groeb_1.html#E2:57"><i><font color="Green" title="E43">A2</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="E11:57"/>
 <a href="termord.html#K5" title="TERMORD:func.5">HM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:57"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="termord.html#D8" title="TERMORD:def.8">TERMORD:def 8</a></i>;<br/>

<b>then </b> <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font> = 
 <a href="polynom7.html#K3" title="POLYNOM7:func.3">coefficient</a> <span class="p1">(<span class="default"><a href="termord.html#K5" title="TERMORD:func.5">HM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9" title="POLYNOM7:th.9">POLYNOM7:9</a></i>
<br/>.= 
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="ref" href="termord.html#T22" title="TERMORD:th.22">TERMORD:22</a></i>
;<br/>
<a NAME="E13:57"/><b>then </b>
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">p</font></span>)</span>,<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>
 
;<br/>
<b>hence </b><a NAME="E14:57"/>
<font color="Maroon" title="c7">c</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c4">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="ref" href="termord.html#T17" title="TERMORD:th.17">TERMORD:17</a></i>;<br/>


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