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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> ;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>
 ;<br/>

<div><i><font color="Green" title="E25">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>thus </b><a NAME="E2:25_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:25_1_1"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 
<b>by </b><i><a class="txt" href="int_7.html#E3:25_1_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">q</font> = <font color="Maroon" title="c2">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c3">q</font> as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <b>by </b><i><a class="txt" href="int_7.html#E3:25_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/>
<a NAME="E7:25_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_7.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="int_7.html#E3:25_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="int_7.html#E10"><i><font color="Green" title="E10">Lm4</font></i></a></i>;<br/>
<a NAME="E8:25_1_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_7.html#E1:25_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T20" title="WSIERP_1:th.20">WSIERP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green" title="E27">A6</font></i>: 
(  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K20" title="RVSUM_1:func.20">Product</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_3.html#D5" title="NAT_3:def.5">NAT_3:def 5</a></i>;<br/><a NAME="E4:25_2"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="valued_0.html#D6" title="VALUED_0:def.6">VALUED_0:def 6</a></i>;<br/><a NAME="E5:25_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_7.html#E3:25_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/><a NAME="E6:25_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="int_7.html#E4:25_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E8:25_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_7.html#E1:25_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="uproots.html#D1" title="UPROOTS:def.1">UPROOTS:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c3">f</font> = 
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="int_7.html#E3:25_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/><b>hence </b><a NAME="E10:25_2"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_7.html#E3:25_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/>
(  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="int_7.html#E2:25"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>


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