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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>









<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>;<br/>
<i><font color="Green">E23</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">u</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1"/>
( <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> or  not <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b3</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:31_2_1_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E4:31_2_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b2</font> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b3</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="polynom1.html#T39">POLYNOM1:39</a></i>;<br/><a NAME="E5:31_2_1_1_1"/><b>then </b>
( <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><a NAME="E6:31_2_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:31_2_1_1_1"/>
<font color="Maroon">b2</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><a NAME="E8:31_2_1_1_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T40">POLYNOM1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> or  not <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_2_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_1_2_1_1_1_1"/>
<font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:31_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T51">POLYNOM1:51</a></i>;<br/><a NAME="E3:31_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T51">POLYNOM1:51</a></i>;<br/><a NAME="E4:31_2_1_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T39">POLYNOM1:39</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_2_1_1_2"/>
not <font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="polyred.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:31_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:31_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">b1</font> <a href="polyred.html#K1" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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