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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>   st <font color="Olive" title="b1">b</font> is <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> &amp; <font color="Maroon" title="c1">a</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="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c1">a</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="Olive" title="b1">b</font></span><br/><b>let </b><font color="Maroon" title="c2">b</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> is <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> &amp; <font color="Maroon" title="c1">a</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="c2">b</font> implies <font color="Maroon" title="c1">a</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="c2">b</font> )</span><br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">b</font> is <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> &amp; <font color="Maroon" title="c1">a</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="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</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="c2">b</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
(  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><a NAME="E1:15_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</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="c2">b</font></span><br/><div class="add"><a NAME="E2:15_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1"/>
<font color="Maroon" title="c1">a</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="c2">b</font>
 <b>by </b><i><a class="txt" href="int_7.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="int_7.html#E1:15_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="wsierp_1.html#T20" title="WSIERP_1:th.20">WSIERP_1:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><a NAME="E1:15_1_2"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</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="c2">b</font></span><br/><div class="add"><b>consider </b><font color="Maroon" title="c3">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="E3:15_1_2"/><i><font color="Green" title="E17">A5</font></i>: 
(  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</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="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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="c2">b</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="E4:15_1_2"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">b</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="E5:15_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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="c2">b</font>
 <b>by </b><i><a class="txt" href="int_7.html#E3:15_1_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/><a NAME="E6:15_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">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#E4:15_1_2"><i><font color="Green" title="E18">A6</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="c4">f</font> = <font color="Maroon" title="c3">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="E8:15_1_2"/>
( <font color="Maroon" title="c2">b</font> is <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c1">a</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="c2">b</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</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="c2">b</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="int_7.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="int_7.html#E1:15_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="int_7.html#E3:15_1_2"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E9:15_1_2"/>
<font color="Maroon" title="c1">a</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="c2">b</font>
 <b>by </b><i><a class="ref" href="int_7.html#T6" target="_self" title="INT_7:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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