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<span class="kw">let </span><font color="Maroon" title="c1">n</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="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">p</font> being 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>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> ) implies <font color="Maroon" title="c1">n</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">p</font> being  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> <span class="kw"> such that </span><a NAME="E2:26"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">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">n</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">r</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="moebius1.html#E5"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E6:26"/>
<font color="Maroon" title="c3">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="wsierp_1.html#T19" title="WSIERP_1:th.19">WSIERP_1:19</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:26"/>
<font color="Maroon" title="c1">n</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</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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