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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be    <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> ;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E35">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E3:50"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">q</font> being  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:50_1"/><i><font color="Green" title="E37">A3</font></i>: 
( <font color="Maroon" title="c3">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="ref" href="pythtrip.html#D2" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a></i>;<br/>
<a NAME="E4:50_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:50_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="pythtrip.html#T7" title="PYTHTRIP:th.7">PYTHTRIP:7</a>, <a class="txt" href="moebius1.html#E3:50_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:50_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:50"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E2:50"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="moebius1.html#T35" target="_self" title="MOEBIUS1:th.35">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T34" target="_self" title="MOEBIUS1:th.34">Th34</a></i>
;<br/>
<b>hence </b><a NAME="E5:50"/>
 <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>


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