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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</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="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">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>







<b>set </b><font color="Maroon" title="c5">ap</font> = <font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font>;<br/>
<i><font color="Green" title="E18">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">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"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></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="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon" title="c6">u</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"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">u'</font> = <font color="Maroon" title="c6">u</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/><span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font>
<b>by </b><i><a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</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="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<i><font color="Green" title="E19">A2</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K13" title="POLYNOM1:func.13">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"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</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="20_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_4"/>
<font color="Maroon" title="c6">u</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"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">u'</font> = <font color="Maroon" title="c6">u</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/><span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</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/>.= 
<font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T40" title="VECTSP_1:th.40">VECTSP_1:40</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font></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"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">u'</font>
<b>by </b><i><a class="ref" href="polynom7.html#D10" title="POLYNOM7:def.10">POLYNOM7:def 10</a></i>
;<br/><b>hence </b><a NAME="E4:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</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="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><a href="polynom1.html#K23" title="POLYNOM1:func.23">-</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="c6">u</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
 <a href="polynom1.html#K23" title="POLYNOM1:func.23">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">a</font> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</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>
 <b>by </b><i><a class="txt" href="polyred.html#E4:20"><i><font color="Green" title="E19">A2</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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