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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">i</font> be   <a href="nat_1.html#NM1">Nat</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">b1</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:156"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:156"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:156"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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>
 ;<br/>

<a NAME="E4:156"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:156"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20" target="_self">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:156"/><i><font color="Green">E46</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" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, <a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:156"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</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">p</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:156"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">b1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b2</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K10" target="_self">-'</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E10:156"/>
<font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, <a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="polynom1.html#T19" target="_self">Th19</a>, <a class="ref" href="polynom1.html#T13" target="_self">Th13</a></i>;<br/>

<b>assume </b><a NAME="E11:156"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b2</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16" target="_self">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E12:156"/>
<font color="Maroon">p</font> <a href="polynom1.html#R3" target="_self">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="polynom1.html#T20" target="_self">Th20</a>, <a class="ref" href="polynom1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E13:156"/><b>then </b>
<font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T21" target="_self">Th21</a>, <a class="ref" href="polynom1.html#T22" target="_self">Th22</a>, </i>;<br/>
<b>hence </b><a NAME="E14:156"/>
contradiction
 <b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self">Def2</a>, <a class="ref" href="polynom1.html#D3" target="_self">Def3</a>, <a class="ref" href="polynom1.html#T19" target="_self">Th19</a>, <a class="ref" href="polynom1.html#T13" target="_self">Th13</a></i>;<br/>


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