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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#V2" title="STRUCT_0:attr.2">empty</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="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>







<b>set </b><font color="Maroon" title="c5">q1</font> = <font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">q2</font> =  <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<i><font color="Green" title="E11">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c7">x</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c8">u</font> or  not <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c8">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1"><b>case </b></a><a NAME="E1:22_2_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c8">u</font>
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b</font></span>)</span>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:22_2_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">u</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E2:22_2_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2"><b>case </b></a><a NAME="E1:22_2_1_1_2"/><i><font color="Green" title="E12">A4</font></i>: 
not <font color="Maroon" title="c4">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c8">u</font>
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_2"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font> <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="c2">L</font>
 <b>by </b><i><a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font> = 
 <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_2.html#E1:22_2_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="polyred.html#D1" title="POLYRED:def.1">POLYRED:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c8">u</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E2:22_2_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:22_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
<font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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