<?xml version="1.0"?>
<div class="add">

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



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  );<br/>
<a NAME="E1:41"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E40</font></i>: 
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:41_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:41_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="real_1.html#K1">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E4:41_1"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1
 
;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E6:41_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i>, <a class="ref" href="int_1.html#T21">INT_1:21</a></i>;<br/>
<a NAME="E7:41_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:41_1"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:41_1"/><b>then </b>
( 2 <a href="series_1.html#K3">to_power</a> 0 <a href="hidden.html#R1">=</a>  <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> &amp; 2 <a href="series_1.html#K3">to_power</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> 0 )
 
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E10:41_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="binari_4.html#K1" target="_self">MajP</a> 2,<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> 2
 
<b>by </b><i/>;<br/>
<i><font color="Green">E50</font></i>: 2 <a href="series_1.html#K3">to_power</a> 2 = 
2 <a href="prepower.html#K3">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="pepin.html#T30">PEPIN:30</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="prepower.html#K3">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
2 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
2 <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/>
<i><font color="Green">E51</font></i>:  <a href="prepower.html#K1">abs</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> 2,<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="prepower.html#K1">abs</a> <span class="p1">(<span class="default">4 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, </i>
<br/>.= 
3
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E13:41_1"/>
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>then </b>3 <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K5">div</a> 1
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
3
<b>by </b><i><a class="ref" href="nat_2.html#T6">NAT_2:6</a></i>
;<br/>
<b>then </b><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default">3 <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K6">mod</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
1 <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/>.= 
1
<b>by </b><i><a class="ref" href="pepin.html#T5">PEPIN:5</a></i>
;<br/>
<a NAME="E16:41_1"/><i><font color="Green">E74</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default">2 <a href="binari_3.html#K1">-BinarySequence</a> 3</span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="binari_4.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> 1,0,<a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="binari_4.html#T2" target="_self">Th2</a>, , <a class="ref" href="binari_3.html#D1">BINARI_3:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font>;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p5">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p4">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="binari_4.html#E1:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
<a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p3">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">h</font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 2,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E19:41_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:41"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">n</font> being 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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>assume </b><a NAME="E2:41_2"/><i><font color="Green">E92</font></i>: 
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 ;<br/>

<b>set </b><font color="Maroon">H1</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font>;<br/>
<b>set </b><font color="Maroon">I1</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font>;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>set </b><font color="Maroon">N</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E3:41_2"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E4:41_2"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E5:41_2"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E6:41_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/>
<a NAME="E7:41_2"/><b>then </b><i><font color="Green">E103</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>
<a NAME="E9:41_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
;<br/>
<a NAME="E10:41_2"/>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">h</font> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E11:41_2"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E12:41_2"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:41_2"/><b>then </b><i><font color="Green">E105</font></i>: 
(  <a href="binari_4.html#K1" target="_self">MajP</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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="nat_1.html#K1">+</a> 1 &amp;  <a href="binari_4.html#K1" target="_self">MajP</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E14:41_2"/><i><font color="Green">E106</font></i>: 
( 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font> &amp; 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E15:41_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">NH</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font>, <font color="Maroon">NI</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</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="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E17:41_2"/><i><font color="Green">E109</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NH</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NI</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E18:41_2"/><i><font color="Green">E110</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">NH</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binari_4.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NH</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NH</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, <a class="ref" href="binari_4.html#T2" target="_self">Th2</a>, <a class="txt" href="binari_4.html#E1:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binari_3.html#T30">BINARI_3:30</a></i>
;<br/>
<i><font color="Green">E112</font></i>: <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">NI</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binari_4.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NI</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <font color="Maroon">NI</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, <a class="ref" href="binari_4.html#T2" target="_self">Th2</a>, <a class="txt" href="binari_4.html#E1:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binari_3.html#T30">BINARI_3:30</a></i>
;<br/>
<a NAME="E21:41_2"/><i><font color="Green">E113</font></i>: 
( <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:41_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p5">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p4">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
<a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p3">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E24:41_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K5">carry</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:41"/>
for <font color="Olive">n</font> being 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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, );</i><br/>


</div>
