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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be  non  <a href="xxreal_0.html#NV2" title="XXREAL_0:NV.2">zero</a>   <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="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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> )</span><br/>

<a NAME="E1:88"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
( <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="88_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:88_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<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>
 ; <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="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>  )</span><br/>

<a NAME="E2:88_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#T55" target="_self" title="MOEBIUS1:th.55">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:88_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1"><span class="lab"><font color="Green" title="E57">A2</font></span></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></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="E2:88"/>
( <font color="Maroon" title="c1">k</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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="xboole_0.html#V1" title="XBOOLE_0:attr.1">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>  )
 
<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">hence </span><a NAME="E3:88"/>
( ( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="moebius1.html#E76"><span class="lab"><font color="Green" title="E54">Lm5</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/>


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