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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_5.html#K3" title="NAT_5:func.3">sigma</a> (<font color="Maroon" title="c1">k</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">b</font> = <span class="p1">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><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#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" onmouseover="rs('uproots/D3')" onmouseout="rh()">UPROOTS:def 3</a></span>;<br/>
<a NAME="E4:44"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
;<br/>
<a NAME="E5:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A3</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:44"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1
 
;<br/>
<a NAME="E7:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A4</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="relat_1.html#T57" onmouseover="rs('relat_1/T57')" onmouseout="rh()">RELAT_1:57</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
1 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c1">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D1" target="_self" onmouseover="rs('nat_5/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
1

;<br/>
<a NAME="E9:44"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#T41" onmouseover="rs('moebius1/T41')" onmouseout="rh()">MOEBIUS1:41</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E10:44"/><span class="kw">then </span>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default">1</span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="finseq_1.html#T94" onmouseover="rs('finseq_1/T94')" onmouseout="rh()">FINSEQ_1:94</a></span>
<br/>.= 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:44"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="finseq_2.html#T34" onmouseover="rs('finseq_2/T34')" onmouseout="rh()">FINSEQ_2:34</a></span>
;<br/>
<a NAME="E12:44"/><span class="kw">then </span>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_5.html#K2" title="NAT_5:func.2">EXP</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="rvsum_1.html#T73" onmouseover="rs('rvsum_1/T73')" onmouseout="rh()">RVSUM_1:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:44"/>
 <a href="nat_5.html#K3" title="NAT_5:func.3">sigma</a> (<font color="Maroon" title="c1">k</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D2" target="_self" onmouseover="rs('nat_5/D2')" onmouseout="rh()">Def2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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