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<div class="add">

<b>set </b><font color="Maroon" title="c1">f</font> =  <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1;<br/>
<b>set </b><font color="Maroon" title="c2">g</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0;<br/>
<a NAME="E1:66"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b>
<font color="Maroon" title="c3">z</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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">z</font> = <font color="Maroon" title="c3">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:66_1"/>
 <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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E5:66_1"/><b>then </b>
not <font color="Maroon" title="c4">z</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="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:66_1"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:66"/><b>then </b><i><font color="Green" title="E41">A1</font></i>: 
 <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></i>;<br/>
<a NAME="E3:66"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:66"/>
 <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 <b>by </b><i><a class="txt" href="moebius1.html#E2:66"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="polynom1.html#D15" title="POLYNOM1:def.15">POLYNOM1:def 15</a></i>;<br/>


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