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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">L</font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</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">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>









<b>set </b><font color="Maroon">q</font> = <font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><i><font color="Green">E23</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></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"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></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="35_1_1_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="E1:35_1_1_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"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">s</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/><b>consider </b><font color="Maroon">t</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:35_1_1_2"/><i><font color="Green">E24</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">t</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><a NAME="E5:35_1_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>;<br/><i><font color="Green">E27</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_2_1"/>
( <font color="Maroon">b</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">s</font> or  not <font color="Maroon">b</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/><i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">s</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:35_1_1_2_2_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="polynom1.html#T73">POLYNOM1:73</a></i>;<br/><a NAME="E7:35_1_1_2_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E9:35_1_1_2_2_1_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E55</font></i>: <font color="Maroon">b1</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="txt" href="polyred.html#E3"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><i><font color="Green">E56</font></i>: <font color="Maroon">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i>, <a class="txt" href="polyred.html#E3"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">s'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:35_1_1_2_2_1_1_4"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">i'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 ;<br/><b>then consider </b><font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_1_4"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b2'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font></span>)</span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">h1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">h2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:35_1_1_2_2_1_1_4"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">h1</font>,<font color="Maroon">h2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">h2</font> )
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/><i><font color="Green">E60</font></i>: <font color="Maroon">h1</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b2'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">b1'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E7:35_1_1_2_2_1_1_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_2_2_1_1_4_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:35_1_1_2_2_1_1_4_2"/><b>then </b>
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1"><i><font color="Green">E38</font></i></a>, , , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:35_1_1_2_2_1_1_4_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:35_1_1_2_2_1_1_4"/>
<font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="binom.html#T1">BINOM:1</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="polyred.html#E1:35_1_1_2_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polyred.html#E3"><i><font color="Green">Lemma39</font></i></a>, <a class="txt" href="polyred.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="polyred.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K3">coefficient</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/><b>hence </b><a NAME="E14:35_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">b</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">s</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="binom.html#T2">BINOM:2</a></i>
;<br/><b>consider </b><font color="Maroon">t</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:35_1_1_2_2_1_2"/><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">t</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:35_1_1_2_2_1_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/><b>then consider </b><font color="Maroon">b1'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2'</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_2_2"/><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b2'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2'</font></span>)</span> )
 <b>by </b><i/>;<br/><i><font color="Green">E69</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2_2_1_2_2_2_1"/>
( <font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> or <font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1_1"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2_2_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">b1'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">h1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">h2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:35_1_1_2_2_1_2_2_2_1_1"/><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">h1</font>,<font color="Maroon">h2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">h2</font> )
 <b>by </b><i>, <a class="txt" href="polyred.html#E3"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/><font color="Maroon">h1</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<font color="Maroon">b2'</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="txt" href="polyred.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">b1'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><a NAME="E5:35_1_1_2_2_1_2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, , <a class="txt" href="polyred.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="polyred.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="termord.html#T1">TERMORD:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2_1_2_2_2_1_2"><b>case </b></a><a NAME="E1:35_1_1_2_2_1_2_2_2_1_2"/>
<font color="Maroon">b1'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_2_2_1_2_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1'</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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_2_2_1_2_2"/>
<font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="binom.html#T1">BINOM:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:35_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="matrlin.html#T15">MATRLIN:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:35_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></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="E5:35_1_1"/>
<font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><i><font color="Green">E83</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></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="35_1_2_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="E1:35_1_2_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"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</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/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">u'</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="binom.html#T1">BINOM:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
;<br/><b>hence </b><a NAME="E4:35_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/></div><b>end;</b></div><a NAME="E4:35_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><i><font color="Green">E85</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></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="35_1_2_4"><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="E1:35_1_2_4"/>
<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"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_2_4_1_1"/>
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> or <font color="Maroon">u'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1_1"><b>case </b></a><a NAME="E1:35_1_2_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:35_1_2_4_1_1_1"/><b>then </b>
( <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font> &amp;  <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="polynom7.html#T8">POLYNOM7:8</a></i>;<br/><a NAME="E3:35_1_2_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>;<br/><b>hence </b><a NAME="E4:35_1_2_4_1_1_1"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_4_1_1_2"><b>case </b></a><a NAME="E1:35_1_2_4_1_1_2"/>
<font color="Maroon">u'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>
;<br/><b>hence </b><a NAME="E3:35_1_2_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:35_1_2_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</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"><a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/></div><b>end;</b></div><a NAME="E7:35_1_2"/><b>then </b>
 <a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E8:35_1_2"/>
<font color="Maroon">a</font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polyred.html#K1" target="_self">*'</a> <font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="polyred.html#E2"><i><font color="Green">Lemma32</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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