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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:90"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 ;<br/>

<a NAME="E2:90"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> holds <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<a NAME="E1:90_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1"><b>suppose </b></a><a NAME="E1:90_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:90_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/><a NAME="E3:90_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E1:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="moebius1.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E4:90_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="moebius1.html#T61" target="_self">Th61</a></i>;<br/><a NAME="E5:90_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:90_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="moebius1.html#T56" target="_self">Th56</a>, <a class="txt" href="moebius1.html#E2:90_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E7:90_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="moebius1.html#E1:90_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>;<br/><b>hence </b><a NAME="E8:90_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:90_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a>, <a class="txt" href="moebius1.html#E3:90_1_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_2"><b>suppose </b></a><a NAME="E1:90_1_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:90_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:90"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="moebius1.html#T19" target="_self">Th19</a></i>;<br/>


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