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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>

<a NAME="E2:31"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>;<br/>
<a NAME="E3:31"/><b>then </b>
not 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:31"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</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/>
<b>set </b><font color="Maroon">M</font> = 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">J</font> = 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font>;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">l</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E10:31"/><i><font color="Green">E42</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E11:31"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">L</font> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<a NAME="E12:31"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E13:31"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:31"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> 1 <a href="nat_1.html#R1">divides</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T13">NAT_2:13</a></i>;<br/>
<a NAME="E16:31"/><b>then </b>
2 <a href="nat_1.html#R1">divides</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E17:31"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="pepin.html#T6">PEPIN:6</a></i>;<br/>
<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span> <a href="int_1.html#K6">mod</a> 2</span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="int_3.html#T15">INT_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 0</span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
0 <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/>
<font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="int_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="int_3.html#T8">INT_3:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i>, <a class="ref" href="int_3.html#T14">INT_3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="int_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="int_3.html#T13">INT_3:13</a></i>
;<br/>
<b>hence </b><a NAME="E21:31"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2
 <b>by </b><i>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>


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