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<b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</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="c3">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green" title="E21">A28</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b'</font> holds <br/>( <font color="Maroon" title="c5">p1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b'</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; ( for <font color="Olive" title="b2">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">b'</font> holds <br/><font color="Maroon" title="c5">p1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b'</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="c3">L</font> ) )
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green" title="E22">A29</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b1">b'</font> holds <br/>( <font color="Maroon" title="c6">p2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b'</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c2">b</font></span>)</span> &amp; ( for <font color="Olive" title="b2">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st not <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">b'</font> holds <br/><font color="Maroon" title="c6">p2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b2">b'</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="c3">L</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1_1"/>
( <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c7">x</font> or  not <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c7">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:24_1_2_1_1_1_1"/><i><font color="Green" title="E23">A30</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">p1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font> = 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c2">b</font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E1:24_1_2"><i><font color="Green" title="E21">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">p2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="polyred.html#E2:24_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="polyred.html#E1:24_1_2_1_1_1_1"><i><font color="Green" title="E23">A30</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:24_1_2_1_1_1_2"/><i><font color="Green" title="E23">A31</font></i>: 
not <font color="Maroon" title="c2">b</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">p1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font> = 
 <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="txt" href="polyred.html#E1:24_1_2"><i><font color="Green" title="E21">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">p2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="polyred.html#E2:24_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="polyred.html#E1:24_1_2_1_1_1_2"><i><font color="Green" title="E23">A31</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:24_1_2_1"/>
<font color="Maroon" title="c5">p1</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24_1_2"/>
<font color="Maroon" title="c5">p1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">p2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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