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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">h</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h</font> <a href="int_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">i</font> <a href="int_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/>

<b>reconsider </b><font color="Maroon">M</font> = 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">h</font></span>)</span></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E3:32"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:32"/><i><font color="Green">E40</font></i>: 
 <a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</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">x</font> = <font color="Maroon">x</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="E7:32"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</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="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<b>then </b><i><font color="Green">E41</font></i>: <font color="Maroon">M</font> <a href="int_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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">x</font></span>)</span></span>)</span> <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>
<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">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">x</font></span>)</span></span>)</span> <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>
<b>by </b><i><a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">x</font></span>)</span></span>)</span> <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>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">L</font> = 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E10:32"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:32"/><i><font color="Green">E42</font></i>: 
 <a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</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">y</font> = <font color="Maroon">y</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="E14:32"/>
<font color="Maroon">L</font> <a href="hidden.html#R1">=</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="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>: <font color="Maroon">L</font> <a href="int_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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">y</font></span>)</span></span>)</span> <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>
<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">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">y</font></span>)</span></span>)</span> <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>
<b>by </b><i><a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">y</font></span>)</span></span>)</span> <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>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/>
<i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span> <a href="int_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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="int_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>
<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">h</font> <a href="int_1.html#K6">mod</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> <a href="int_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>
<b>by </b><i/>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="int_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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="int_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>
<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">i</font> <a href="int_1.html#K6">mod</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> <a href="int_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>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E18:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font></span>)</span> <a href="int_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><a href="binari_4.html#K1" target="_self">MajP</a> <font color="Maroon">n</font>,<span class="p4">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="int_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>
 <b>by </b><i>, </i>;<br/>


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