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<b>consider </b><font color="Maroon">b11</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b11</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; ( for <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Maroon">b11</font> <a href="polynom1.html#R1">&lt;</a> <font color="Olive">b'</font> holds <br/><font color="Maroon">p</font> <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/>;<br/>
<b>consider </b><font color="Maroon">b22</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b22</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; ( for <font color="Olive">b'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>  st <font color="Maroon">b22</font> <a href="polynom1.html#R1">&lt;</a> <font color="Olive">b'</font> holds <br/><font color="Maroon">q</font> <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/>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font>;<br/>
<b>consider </b><font color="Maroon">s</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="E6:9_1_1"/><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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"><font color="Maroon">p</font> <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">q</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/>
<b>consider </b><font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E8:9_1_1"/><i><font color="Green">E27</font></i>: 
(  <a href="polynom1.html#K20">divisors</a> <span class="p1">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font> &amp; ( for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/>
<a NAME="E9:9_1_1"/>
<font color="Maroon">b11</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T54">POLYNOM1:54</a></i>;<br/>
<a NAME="E10:9_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">b11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">sgm</font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font>;<br/>
<a NAME="E11:9_1_1"/>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:9_1_1"/><b>then </b>
<font color="Maroon">b11</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:9_1_1"/><i><font color="Green">E49</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="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b11</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:9_1_1"/><i><font color="Green">E50</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> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E16:9_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b11</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E17:9_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></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">b11</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b11</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E18:9_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></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">b11</font>,<font color="Maroon">b22</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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 NAME="E20:9_1_1"/><b>then </b><i><font color="Green">E55</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">s</font>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><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="E1:9_1_1_2"/><i><font color="Green">E57</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">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="E3:9_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2'</font> )
 <b>by </b><i>, <a class="ref" href="polynom1.html#T72">POLYNOM1:72</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="E5:9_1_1_2"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</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"><font color="Maroon">p</font> <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">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><i><font color="Green">E60</font></i>: <font color="Maroon">b1</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/><i><font color="Green">E61</font></i>: <font color="Maroon">b2</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> 2
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">b2'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_2_3"/>
( <font color="Maroon">p</font> <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> &amp; <font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</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="E2:9_1_1_2_3"/><b>then </b>
( not <font color="Maroon">b11</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">b1</font> &amp; not <font color="Maroon">b22</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">b2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:9_1_1_2_3"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">b1</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b11</font> &amp; <font color="Maroon">b2</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b22</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_2_3_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b11</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b22</font> )
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b22</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
<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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</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">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></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="E3:9_1_1_2_3_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="9_1_1_2_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_3_2_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b11</font> or <font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b22</font> )
 </span><b>by </b><i><a class="txt" href="polyred.html#E6:9_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_3_2_1_1"><b>case </b></a><a NAME="E1:9_1_1_2_3_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b11</font>
 ;<br/><div class="add"><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_3_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_2_3_2_1_1_1"/>
<font color="Maroon">b1</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font>
 ;<br/><a NAME="E2:9_1_1_2_3_2_1_1_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font></span>)</span> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_2_3_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E8:9_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/></div><b>end;</b></div><a NAME="E3:9_1_1_2_3_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="txt" href="polyred.html#E2:9_1_1"><i><font color="Green">E23</font></i></a>, , </i>;<br/><a NAME="E4:9_1_1_2_3_2_1_1"/>
<font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b2</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:9_1_1_2_3_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="polyred.html#E2:9_1_1"><i><font color="Green">E23</font></i></a>, , <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="9_1_1_2_3_2_1_2"><b>case </b></a><a NAME="E1:9_1_1_2_3_2_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b22</font>
 ;<br/><div class="add"><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_3_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_2_3_2_1_2_1"/>
<font color="Maroon">b2</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b22</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font>
 ;<br/><a NAME="E2:9_1_1_2_3_2_1_2_1"/><b>then </b>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b22</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font></span>)</span> <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/><b>hence </b><a NAME="E3:9_1_1_2_3_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polyred.html#E8:9_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>;<br/></div><b>end;</b></div><a NAME="E3:9_1_1_2_3_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b22</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , <a class="txt" href="polyred.html#E2:9_1_1"><i><font color="Green">E23</font></i></a>, , </i>;<br/><a NAME="E4:9_1_1_2_3_2_1_2"/>
<font color="Maroon">b22</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b1</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:9_1_1_2_3_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="polyred.html#E2:9_1_1"><i><font color="Green">E23</font></i></a>, , <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="E6:9_1_1_2_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_4_1"/>
( <font color="Maroon">p</font> <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> or <font color="Maroon">q</font> <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> )
 </span><b>by </b><i><a class="txt" href="polyred.html#E4:9_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_4_1_1"><b>case </b></a><a NAME="E1:9_1_1_2_4_1_1"/>
<font color="Maroon">p</font> <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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <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">q</font> <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="9_1_1_2_4_1_2"><b>case </b></a><a NAME="E1:9_1_1_2_4_1_2"/>
<font color="Maroon">q</font> <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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_4_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <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">q</font> <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="E10:9_1_1_2"/>
<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>
<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="E24:9_1_1"/><i><font color="Green">E83</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span></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">s</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"><font color="Maroon">p</font> <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">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<i><font color="Green">E84</font></i>: <font color="Maroon">b1</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b22</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">b11</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<font color="Maroon">b2</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b11</font>,<font color="Maroon">b22</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">b22</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E27:9_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b22</font></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>;<br/>
<a NAME="E28:9_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E29:9_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b11</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b22</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/>
<a NAME="E30:9_1_1"/><b>then </b>
<font color="Maroon">b11</font> <a href="polynom1.html#K9">+</a> <font color="Maroon">b22</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E31:9_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="polynom7.html#T1">POLYNOM7:1</a></i>;<br/>
<b>hence </b><a NAME="E32:9_1_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">q</font> is <a href="polynom7.html#V1">non-zero</a>
 <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/>


</div>
