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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="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <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="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">a'</font> be  <a href="rlvect_1.html#V8">non-zero</a> <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>set </b><font color="Maroon">m1</font> =  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">m2</font> =  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">m3</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> <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/>
<a NAME="E1:8"/><i><font color="Green">E32</font></i>: 
( <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> &amp; <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> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<a NAME="E2:8"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <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>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font> is <a href="rlvect_1.html#V8">non-zero</a>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<i><font color="Green">E35</font></i>: <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">b</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#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</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>
<b>by </b><i><a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/>
<i><font color="Green">E37</font></i>: <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">b'</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#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</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>
<b>by </b><i><a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:8_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:8_3"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> holds <br/><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="Olive">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><a class="ref" href="polynom7.html#D4">POLYNOM7:def 4</a></i>;<br/><b>assume </b><a NAME="E4:8_3"/>
<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#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E5:8_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">u</font> <a href="pboole.html#R6">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E6:8_3"/>
<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">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:8_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>assume </b><a NAME="E1:8_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 ;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:8_4"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Olive">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> holds <br/><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="Olive">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><a class="ref" href="polynom7.html#D4">POLYNOM7:def 4</a></i>;<br/><b>assume </b><a NAME="E4:8_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="polynom1.html#K15">.</a> <font color="Maroon">u</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="E5:8_4"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">u</font> <a href="pboole.html#R6">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><a NAME="E6:8_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="polynom1.html#K15">.</a> <font color="Maroon">b'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:8_4"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:8"/><i><font color="Green">E52</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> <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="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</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> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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">s</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> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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">s</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"><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">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/>
<i><font color="Green">E53</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</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> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>set </b><font color="Maroon">u</font> = <font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font>;<br/>
<b>consider </b><font color="Maroon">k'</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="E12:8"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">k'</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> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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">b</font>,<font color="Maroon">b'</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/>
<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="E14:8"/><i><font color="Green">E55</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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">s</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"><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">b2</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>
<i><font color="Green">E57</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">b'</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">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E58</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">b'</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="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/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:8_8"/><i><font color="Green">E60</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">s</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k'</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="E3:8_8"/><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></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">s</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"><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">b2</font></span>)</span> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_8_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b'</font> )
 ;<br/><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <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> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</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="E3:8_8_1"/>
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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_8_2_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> or <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_8_2_1_1"><b>case </b></a><a NAME="E1:8_8_2_1_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:8_8_2_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">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/><b>hence </b><a NAME="E3:8_8_2_1_1"/>
<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"><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">b2</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="binom.html#T1">BINOM:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_8_2_1_2"><b>case </b></a><a NAME="E1:8_8_2_1_2"/>
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 ;<br/><div class="add"><a NAME="E2:8_8_2_1_2"/><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">b2</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/><b>hence </b><a NAME="E3:8_8_2_1_2"/>
<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"><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">b2</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="binom.html#T2">BINOM:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:8_8"/>
<font color="Maroon">s</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/>;<br/></div><b>end;</b></div>
<a NAME="E18:8"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font>
 
<b>by </b><i>, , <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>;<br/>
<a NAME="E19:8"/><b>then </b><i><font color="Green">E64</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> <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="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , , , , , , <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>then </b><i><font color="Green">E65</font></i>:  <a href="polynom7.html#K2">term</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> <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> = 
<font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font>
<b>by </b><i><a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
<br/>.= 
 <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
;<br/>
<i><font color="Green">E78</font></i>:  <a href="polynom7.html#K3">coefficient</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> <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> = 
<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> <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="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<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> <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="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font>
<b>by </b><i>, , , , , , , , , <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
 <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</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>thus </b> <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span> = 
 <a href="polynom7.html#K1">Monom</a> <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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
<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> <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>
<b>by </b><i>, , <a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
;<br/>


</div>
