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<b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">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">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; not <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> ) or ( not <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> ) or ( <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; not <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> ) or ( <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">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">E42</font></i>: 
( not <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; not <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E3:49_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T47">NAT_3:47</a>, </i>;<br/><i><font color="Green">E56</font></i>:  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>
;<br/><a NAME="E5:49_1_1"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">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">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:49_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:49_1_1_2"/>
( <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> or <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="txt" href="moebius1.html#E1:49"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:49_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1"><i><font color="Green">Lemma36</font></i></a>, </i>;<br/>


</div><b>end;</b></div> <a href="moebius1.html#K2" target="_self">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/><b>hence </b><a NAME="E7:49_1_1"/>
 <a href="moebius1.html#K2" target="_self">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><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">E74</font></i>: 
( not <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="moebius1.html#E1"><i><font color="Green">Lemma36</font></i></a></i>;<br/><a NAME="E5:49_1_2"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a>, </i>;<br/><a NAME="E6:49_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:49_1_2"/>
 <a href="moebius1.html#K2" target="_self">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, </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">E81</font></i>: 
( <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; not <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_3"/><b>then </b><i><font color="Green">E91</font></i>: 
(  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="moebius1.html#E1"><i><font color="Green">Lemma36</font></i></a></i>;<br/><a NAME="E5:49_1_3"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a>, </i>;<br/><a NAME="E6:49_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:49_1_3"/>
 <a href="moebius1.html#K2" target="_self">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, </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">E93</font></i>: 
( <font color="Maroon">a</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> &amp; <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 ;<br/><div class="add"><a NAME="E2:49_1_4"/><b>then </b><i><font color="Green">E94</font></i>: 
(  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E4:49_1_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1"><i><font color="Green">Lemma36</font></i></a>, </i>;<br/><a NAME="E5:49_1_4"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T56">NAT_1:56</a>, </i>;<br/><a NAME="E6:49_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:49_1_4"/>
 <a href="moebius1.html#K2" target="_self">Moebius</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
