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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">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> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( ( not <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; not <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) or ( not <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) or ( <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; not <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) or ( <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
( not <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; not <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_1"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
(  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/><a NAME="E3:49_1_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:49"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="nat_3.html#T47" title="NAT_3:th.47">NAT_3:47</a></i>;<br/><b>then </b><i><font color="Green" title="E38">A4</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>
;<br/><a NAME="E5:49_1_1"/>
not <font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E3:49_1_1_2"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c3">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">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/>
<a NAME="E4:49_1_1_2"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:49_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">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">a</font> or <font color="Maroon" title="c3">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="c2">b</font> )
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:49"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="moebius1.html#E3:49_1_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="moebius1.html#T11" target="_self" title="MOEBIUS1:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E6:49_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:49_1_1"><i><font color="Green" title="E36">A2</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="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E4:49_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
;<br/><b>hence </b><a NAME="E7:49_1_1"/>
 <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">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="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:49_1_1"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green" title="E36">A6</font></i>: 
( not <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_2"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
(  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_2"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c3">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="c2">b</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:49_1_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><a NAME="E5:49_1_2"/>
<font color="Maroon" title="c3">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">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:49_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E6:49_1_2"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 <b>by </b><i><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:49_1_2"/>
 <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">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="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:49_1_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_3"><b>suppose </b></a><a NAME="E1:49_1_3"/><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; not <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_3"/><b>then </b><i><font color="Green" title="E37">A10</font></i>: 
(  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_3"/><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c3">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">a</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:49_1_3"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><a NAME="E5:49_1_3"/>
<font color="Maroon" title="c3">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">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:49_1_3"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E6:49_1_3"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 <b>by </b><i><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:49_1_3"/>
 <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">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="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:49_1_3"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_4"><b>suppose </b></a><a NAME="E1:49_1_4"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c1">a</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> &amp; <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_4"/><b>then </b><i><font color="Green" title="E37">A13</font></i>: 
(  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_4"/><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c3">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">a</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:49_1_4"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><a NAME="E5:49_1_4"/>
<font color="Maroon" title="c3">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">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:49_1_4"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E6:49_1_4"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 <b>by </b><i><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:49_1_4"/>
 <a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" title="MOEBIUS1:func.2">Moebius</a> <font color="Maroon" title="c1">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="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:49_1_4"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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