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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> <font color="Maroon" title="c1">k</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a> </span><br/>

<a NAME="E1:53"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">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> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">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> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> = 
 <a href="nat_5.html#K3" title="NAT_5:func.3">sigma</a> (<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D3" target="_self" title="NAT_5:def.3">Def3</a></span>
<br/>.= 
 <a href="uproots.html#K3" title="UPROOTS:func.3">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> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#D2" target="_self" title="NAT_5:def.2">Def2</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>
<span class="kw">hence </span><a NAME="E2:53"/>
 <a href="nat_5.html#K4" title="NAT_5:func.4">Sigma</a> <font color="Maroon" title="c1">k</font> is  <a href="nat_5.html#V1" title="NAT_5:attr.1">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T34" target="_self" title="NAT_5:th.34">Th34</a>, <a class="ref" href="nat_5.html#T35" target="_self" title="NAT_5:th.35">Th35</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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