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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="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</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>, <font color="Maroon" title="c4">q</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/>





<i><font color="Green" title="E8">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#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</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"><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> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</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="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon" title="c5">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"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">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/><span class="p1">(<span class="default"><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> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><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> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</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"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">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/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T45" title="RLVECT_1:th.45">RLVECT_1:45</a></i>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</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="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></i>
;<br/><b>hence </b><a NAME="E4:10_2"/>
<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="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
 <a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</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#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">-</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E1:10"><i><font color="Green" title="E8">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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