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<span class="kw">set </span><font color="Maroon" title="c1">f</font> =  <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1;<br/>
<a NAME="E1:65"/>
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#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> 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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z</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> 1</span>)</span> implies <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c2">z</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> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">z</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="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">z</font> = <font color="Maroon" title="c2">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="E4:65_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="E5:65_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">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>
 
<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="E6:65_1"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">z</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="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <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#NK2" title="RELAT_1:NK.2">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="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T17" title="FUNCOP_1:th.17">FUNCOP_1:17</a></span>;<br/>
<a NAME="E3:65"/>
 <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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:65"/>
 <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="pre_poly.html#D13" title="PRE_POLY:def.13">PRE_POLY:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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