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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/>
<font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><a NAME="E2:1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="abian.html#D2">ABIAN:def 2</a></i>;<br/><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
1 <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="power.html#T54">POWER:54</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="power.html#T31">POWER:31</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/>
not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 ;<br/><div class="add"><a NAME="E2:1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_2.html#T24">NAT_2:24</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:1_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default">1 <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T55">POWER:55</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T31">POWER:31</a></i>
;<br/><a NAME="E8:1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E9:1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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