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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<a NAME="E1:89"/><i><font color="Green" title="E56">A1</font></i>: 
( <font color="Maroon" title="c1">k</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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>  &amp; <font color="Maroon" title="c2">n</font> is  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:89"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c2">n</font> implies ( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; not <font color="Maroon" title="c1">k</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c2">n</font>
 ;<br/>

<a NAME="E2:89_1"/>
 <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="moebius1.html#T55" target="_self" title="MOEBIUS1:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E3:89_1"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; not <font color="Maroon" title="c1">k</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 <b>by </b><i><a class="txt" href="moebius1.html#E1:89_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="moebius1.html#T26" target="_self" title="MOEBIUS1:th.26">Th26</a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:89"/>
( ( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; not <font color="Maroon" title="c1">k</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) iff <font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="moebius1.html#E1:89"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="moebius1.html#E76"><i><font color="Green" title="E54">Lm5</font></i></a></i>;<br/>


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