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<a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:64_1_1_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:64_1_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:64_1_1_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_1_1"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n</font></span>)</span> or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><a NAME="E2:64_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1_1_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_2"/>
 not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/><div class="add"><a NAME="E2:64_1_1_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></span>;<br/><a NAME="E3:64_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:64_1_1"/>
 <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:64_1_1"/>
(  <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font> is  <a href="pre_poly.html#V2" title="PRE_POLY:attr.2">finite-support</a>  &amp;  <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">n</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="pre_poly.html#D8" title="PRE_POLY:def.8">PRE_POLY:def 8</a>, <a class="ref" href="valued_0.html#D6" title="VALUED_0:def.6">VALUED_0:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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