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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_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="E1:48_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="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
( 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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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></span><br/><div class="add"><a NAME="E2:48_1_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
not <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="48_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:48_1_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</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="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</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="E4:48_1_1_1"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E5:48_1_1_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E3:48_1_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="moebius1.html#T11" target="_self" title="MOEBIUS1:th.11">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:48_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</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><a NAME="E3:48_1_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
 <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="real_1.html#K1" title="REAL_1:func.1">-</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></span>;<br/><a NAME="E4:48_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="nat_1.html#K4" title="NAT_1:func.4">*</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="nat_1.html#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="nat_3.html#T47" title="NAT_3:th.47">NAT_3:47</a></span>;<br/><span class="kw">then </span> <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="nat_1.html#K4" title="NAT_1:func.4">*</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="nat_1.html#K2" title="NAT_1:func.2">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></span>
<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="nat_1.html#K2" title="NAT_1:func.2">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></span>
<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="nat_1.html#K2" title="NAT_1:func.2">+</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></span>
;<br/><span class="kw">then </span> <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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="nat_1.html#K2" title="NAT_1:func.2">+</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:48_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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:48_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</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="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
( 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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_2"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<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>
 <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="E4:48_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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/><a NAME="E5:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 <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="E6:48_1_2"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</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="48_1_3"><span class="kw">suppose </span></a><a NAME="E1:48_1_3"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_3"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<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>
 <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="E4:48_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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_3"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/><a NAME="E5:48_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 <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="E6:48_1_3"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_3"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_3"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</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="48_1_4"><span class="kw">suppose </span></a><a NAME="E1:48_1_4"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:48_1_4"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<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>
 <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="E4:48_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="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_4"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/><a NAME="E5:48_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font> is  <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a> 
 <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="E6:48_1_4"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_4"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_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="nat_1.html#K4" title="NAT_1:func.4">*</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="binop_2.html#K11" title="BINOP_2:func.11">*</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_4"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="moebius1.html#D3" target="_self" title="MOEBIUS1:def.3">Def3</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>

</div>
