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<a NAME="E1:42_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font> being   <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>  holds <br/> ( ( not <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n2</font> ) &amp; ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b1">n1</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="b3">m</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive" title="b2">n2</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="b3">m</font></span>)</span></span>)</span> ) implies <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n1</font>, <font color="Maroon" title="c4">n2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> ) &amp; ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">n1</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">m</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c4">n2</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">m</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:42_1_1_1"/>
( not <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c4">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> )
 ; <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">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">n1</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">m</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c4">n2</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">m</font></span>)</span></span>)</span> ) implies <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:42_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; not <font color="Maroon" title="c3">n1</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">m</font></span>)</span></span>)</span> ) or  ex <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; not <font color="Maroon" title="c4">n2</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">m</font></span>)</span></span>)</span> ) or <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c2">n</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<span class="kw">assume </span><a NAME="E4:42_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c3">n1</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">m</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; not <font color="Maroon" title="c4">n2</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">m</font></span>)</span></span>)</span> ) or <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font> )</span><br/>

<a NAME="E5:42_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c3">n1</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="c5">m</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">assume </span><a NAME="E6:42_1_1_1"/>
for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c4">n2</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">m</font></span>)</span></span>)</span>
 ; <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="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font></span><br/>

<span class="kw">hence </span><a NAME="E7:42_1_1_1"/>
<font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></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="E2:42_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  ex <font color="Olive" title="b1">n1</font> being   <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>  st <br/>for <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n1</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="b2">m</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font> being   <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>  st <br/>for <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n1</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="b2">m</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">n9</font> = <font color="Maroon" title="c2">n</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">n1</font> =  <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="c3">n9</font></span>)</span></span>)</span> 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">n1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c4">n1</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">m</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">m</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> implies <font color="Maroon" title="c4">n1</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="c5">m</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:42_1_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 ; <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="c4">n1</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="c5">m</font></span>)</span></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E5:42_1_1_2"/>
<font color="Maroon" title="c4">n1</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="c5">m</font></span>)</span></span>)</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>
<span class="kw">hence </span><a NAME="E3:42_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <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>  holds  verum ) &amp; ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <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>  st <br/>for <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive">b<sub>1</sub></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="b2">m</font></span>)</span></span>)</span> ) &amp; ( not <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being   <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>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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>  holds <br/> ( ( <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive">b<sub>1</sub></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="b3">m</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Olive">b<sub>2</sub></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="b3">m</font></span>)</span></span>)</span> ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( not <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></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>
