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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</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="c1">p</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> </span><br/>

<span class="kw">assume </span><a NAME="E1:30"/>
<font color="Maroon" title="c1">p</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">n</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">n</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">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></span>;<br/>
<a NAME="E4:30"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<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> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T3" title="NAT_3:th.3">NAT_3:3</a></span>;<br/>
<a NAME="E5:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:30_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></span>;<br/><a NAME="E3:30_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E4:30_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T31" title="PEPIN:th.31">PEPIN:31</a></span>;<br/><span class="kw">hence </span><a NAME="E5:30_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</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></div><span class="kw">end;</span></div>

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