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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:155"/><i><font color="Green" title="E86">A1</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:155"/><i><font color="Green" title="E87">A2</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <b>and </b><br/><a NAME="E3:155"/><i><font color="Green" title="E88">A3</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ;<br/>

<a NAME="E4:155"/><i><font color="Green" title="E89">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>
<a NAME="E5:155"/><b>then </b><i><font color="Green" title="E90">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">divisors</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c2">i</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" target="_self" title="POLYNOM1:def.14">Def14</a></i>;<br/>
<a NAME="E7:155"/><i><font color="Green" title="E91">A6</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E1:155"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="polynom1.html#E2:155"><i><font color="Green" title="E87">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:155"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" title="POLYNOM1:func.21">decomp</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">/.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">p</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19" target="_self" title="POLYNOM1:def.19">Def19</a></i>;<br/>
<a NAME="E9:155"/><b>then </b><i><font color="Green" title="E92">A7</font></i>: 
( <font color="Maroon" title="c4">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <font color="Maroon" title="c6">p</font> )
 
<b>by </b><i><a class="txt" href="polynom1.html#E3:155"><i><font color="Green" title="E88">A3</font></i></a>, <a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/>
<b>hence </b><a NAME="E10:155"/>
<font color="Maroon" title="c4">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E1:155"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="polynom1.html#E2:155"><i><font color="Green" title="E87">A2</font></i></a>, <a class="txt" href="polynom1.html#E5:155"><i><font color="Green" title="E90">A5</font></i></a>, <a class="ref" href="polynom1.html#T70" target="_self" title="POLYNOM1:th.70">Th70</a></i>;<br/>

<b>assume </b><a NAME="E11:155"/><i><font color="Green" title="E93">A8</font></i>: 
<font color="Maroon" title="c5">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E12:155"/>
<font color="Maroon" title="c6">p</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E4:155"><i><font color="Green" title="E89">A4</font></i></a>, <a class="txt" href="polynom1.html#E7:155"><i><font color="Green" title="E91">A6</font></i></a>, <a class="ref" href="polynom1.html#T68" target="_self" title="POLYNOM1:th.68">Th68</a></i>;<br/>
<a NAME="E13:155"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="polynom1.html#E9:155"><i><font color="Green" title="E92">A7</font></i></a>, <a class="txt" href="polynom1.html#E11:155"><i><font color="Green" title="E93">A8</font></i></a>, <a class="ref" href="polynom1.html#T61" target="_self" title="POLYNOM1:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E14:155"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E1:155"><i><font color="Green" title="E86">A1</font></i></a>, <a class="txt" href="polynom1.html#E2:155"><i><font color="Green" title="E87">A2</font></i></a>, <a class="txt" href="polynom1.html#E5:155"><i><font color="Green" title="E90">A5</font></i></a>, <a class="ref" href="polynom1.html#T70" target="_self" title="POLYNOM1:th.70">Th70</a></i>;<br/>


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