<?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>[ 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> ] <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> <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; <font color="Maroon">h</font> <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">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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> &amp; <font color="Maroon">i</font> <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">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; ( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) ) implies  <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> );<br/>
<a NAME="E1:44"/><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="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_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> <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; <font color="Maroon">h</font> <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">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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> &amp; <font color="Maroon">i</font> <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">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; ( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/>

1 <a href="binarith.html#K3">-'</a> 1 = 
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/>.= 
0

;<br/>
<a NAME="E3:44_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <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="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 1 )
 
<b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="binari_4.html#K2" target="_self">2sComplement</a> 1,0 = 
1 <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="binari_3.html#K1">-BinarySequence</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> 1
<b>by </b><i><a class="ref" href="binari_3.html#T26">BINARI_3:26</a></i>
<br/>.= 
1 <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>
;<br/>
<a NAME="E5:44_1"/><i><font color="Green">E43</font></i>: 
( 2 <a href="series_1.html#K3">to_power</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> = 
 <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/>.= 
1

;<br/>
<a NAME="E7:44_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="binari_4.html#K1" target="_self">MajP</a> 1,<span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E45</font></i>:  <a href="euclid.html#K5">0*</a> 0 = 
0 <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/>
<i><font color="Green">E50</font></i>:  <a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> = 
1 <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1 <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
1 <a href="binari_3.html#K1">-BinarySequence</a> 1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> 0</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="binari_3.html#T29">BINARI_3:29</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_6_1"/>
( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_6_1_1"><b>suppose </b></a><a NAME="E1:44_1_6_1_1"/>
( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:44_1_6_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 &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="E3:44_1_6_1_1"/><b>then </b>
<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/><b>hence </b><a NAME="E4:44_1_6_1_1"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="binari_4.html#T1" target="_self">Th1</a>, , <a class="ref" href="binari_3.html#T18">BINARI_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_6_1_2"><b>suppose </b></a><a NAME="E1:44_1_6_1_2"/><i><font color="Green">E52</font></i>: 
( <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#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:44_1_6_1_2"/><b>then </b>
<font color="Maroon">h</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="E3:44_1_6_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">h</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="E4:44_1_6_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="binari_4.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:44_1_6_1_2"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="binari_4.html#T2" target="_self">Th2</a>, , <a class="ref" href="binari_3.html#T18">BINARI_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:44_1"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> 1,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/><i><font color="Green">E75</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="44_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="E1:44_2"/>
(  <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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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">n</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> &amp; <font color="Maroon">i</font> <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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &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">n</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> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; ( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) ) implies  <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:44_2"/><i><font color="Green">E91</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"><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; <font color="Maroon">h</font> <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"><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="real_1.html#K5">-</a> 1 &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> &amp; <font color="Maroon">i</font> <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"><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="real_1.html#K5">-</a> 1 &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> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <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"><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="real_1.html#K5">-</a> 1 &amp; ( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) ) )
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<font color="Maroon">h</font>;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<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"><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">I1</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">n2</font> = 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="margrel1.html#K7">FALSE</a> ;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="margrel1.html#K8">TRUE</a> ;<br/>
<a NAME="E3:44_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="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
;<br/>
<a NAME="E4:44_2"/><b>then </b><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> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</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="real_1.html#K5">-</a> 1 &amp;  <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">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</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="real_1.html#K5">-</a> 1 &amp;  <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">n</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="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</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="real_1.html#K5">-</a> 1 )
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E6:44_2"/><i><font color="Green">E93</font></i>: 
 <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">a</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> <b> such that </b><br/><a NAME="E8:44_2"/><i><font color="Green">E94</font></i>: 
 <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" target="_self">2sComplement</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_1_1"/>
( ( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) or ( <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#NR2" target="_self">&gt;=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_2_1_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">h</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</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/><i><font color="Green">E103</font></i>:  <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> = 
<span class="p1">(<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="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<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">h</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E4:44_2_1_1_1"/><i><font color="Green">E104</font></i>: 
(  <a href="finseq_1.html#K3">len</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="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"><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="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 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="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:44_2_1_1_1"/>
<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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E6:44_2_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:44_2_1_1_1"/><b>then </b><i><font color="Green">E106</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="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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="binari_3.html#T27">BINARI_3:27</a></i>;<br/><a NAME="E8:44_2_1_1_1"/>
 <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="E9:44_2_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="prepower.html#K1">abs</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:44_2_1_1_1"/>
<font color="Maroon">n</font> <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#T31">XREAL_1:31</a></i>;<br/><a NAME="E11:44_2_1_1_1"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</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="power.html#T44">POWER:44</a></i>;<br/><a NAME="E12:44_2_1_1_1"/><b>then </b>
 <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="txt" href="binari_4.html#E1:44"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:44_2_1_1_1"/><b>then </b><i><font color="Green">E110</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">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="E14:44_2_1_1_1"/><i><font color="Green">E111</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">i</font> <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="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="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><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <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">n</font></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> <font color="Maroon">n</font>

;<br/><a NAME="E16:44_2_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <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">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/><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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</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> = 
<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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p5">(<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></span>)</span></span>)</span></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>, , </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="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, , <a class="ref" href="binari_3.html#T30">BINARI_3:30</a></i>
;<br/><a NAME="E19:44_2_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <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#T10">XREAL_1:10</a></i>;<br/><a NAME="E20:44_2_1_1_1"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</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="power.html#T44">POWER:44</a></i>;<br/><a NAME="E21:44_2_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="binari_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:44_2_1_1_1"/><i><font color="Green">E116</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> <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="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="binari_4.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green">E117</font></i>:  <a href="binari_2.html#K4">Intval</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="binarith.html#K7">Absval</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>
<b>by </b><i>, <a class="ref" href="binari_2.html#D3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i>, , <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
;<br/><i><font color="Green">E118</font></i>:  <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> = 
<span class="p1">(<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="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><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">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></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<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>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E25:44_2_1_1_1"/><i><font color="Green">E119</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> <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="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="binari_4.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green">E120</font></i>:  <a href="binari_2.html#K4">Intval</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> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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="binari_2.html#D3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon">NI</font> <a href="xcmplx_0.html#K6">-</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="binari_4.html#T5" target="_self">Th5</a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon">i</font>

;<br/><i><font color="Green">E121</font></i>:  <a href="binari_2.html#K5">Int_add_ovfl</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>,<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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</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"><font color="Maroon">n</font> <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"><font color="Maroon">n</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="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="binari_2.html#D4">BINARI_2:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/> <a href="binari_2.html#K6">Int_add_udfl</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>,<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> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</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"><font color="Maroon">n</font> <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"><font color="Maroon">n</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="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="binari_2.html#D5">BINARI_2:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/><b>then </b> <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</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="binari_4.html#T4" target="_self">Th4</a>, , , <a class="ref" href="binari_2.html#T14">BINARI_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0

;<br/><b>hence </b><a NAME="E30:44_2_1_1_1"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_1_1_2"/><i><font color="Green">E122</font></i>: 
( <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#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i</font> = <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/><i><font color="Green">E123</font></i>:  <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> = 
<span class="p1">(<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="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<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">i</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E4:44_2_1_1_2"/><i><font color="Green">E124</font></i>: 
(  <a href="finseq_1.html#K3">len</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="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"><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="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 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="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:44_2_1_1_2"/>
<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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E6:44_2_1_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:44_2_1_1_2"/><b>then </b><i><font color="Green">E126</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="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> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="binari_3.html#T27">BINARI_3:27</a></i>;<br/><a NAME="E8:44_2_1_1_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>
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:44_2_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
 <a href="prepower.html#K1">abs</a> <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E10:44_2_1_1_2"/>
<font color="Maroon">n</font> <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#T31">XREAL_1:31</a></i>;<br/><a NAME="E11:44_2_1_1_2"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</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="power.html#T44">POWER:44</a></i>;<br/><a NAME="E12:44_2_1_1_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>
 <b>by </b><i><a class="txt" href="binari_4.html#E1:44"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:44_2_1_1_2"/><b>then </b><i><font color="Green">E128</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
 <b>by </b><i/>;<br/><a NAME="E14:44_2_1_1_2"/><i><font color="Green">E129</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> <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="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</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> <font color="Maroon">n</font></span>)</span></span>)</span> = 
<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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="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><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<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> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <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">n</font></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> <font color="Maroon">n</font>

;<br/><a NAME="E16:44_2_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <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>
 <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> 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/><i><font color="Green">E131</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="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> = 
<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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p5">(<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></span>)</span></span>)</span></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>, , </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="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, , <a class="ref" href="binari_3.html#T30">BINARI_3:30</a></i>
;<br/><a NAME="E19:44_2_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <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#T10">XREAL_1:10</a></i>;<br/><a NAME="E20:44_2_1_1_2"/><b>then </b>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</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="power.html#T44">POWER:44</a></i>;<br/><a NAME="E21:44_2_1_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<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="binari_4.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:44_2_1_1_2"/><i><font color="Green">E133</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> <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="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="binari_4.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green">E134</font></i>:  <a href="binari_2.html#K4">Intval</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="binarith.html#K7">Absval</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>
<b>by </b><i>, <a class="ref" href="binari_2.html#D3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
;<br/><i><font color="Green">E135</font></i>:  <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> = 
<span class="p1">(<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="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <span class="p2">(<span class="default"><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">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></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<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>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E25:44_2_1_1_2"/><i><font color="Green">E136</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> <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="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="binari_4.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green">E137</font></i>:  <a href="binari_2.html#K4">Intval</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> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">Absval</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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="binari_2.html#D3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon">NH</font> <a href="xcmplx_0.html#K6">-</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="binari_4.html#T5" target="_self">Th5</a>, <a class="ref" href="binari_3.html#T36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon">h</font>

;<br/><i><font color="Green">E138</font></i>:  <a href="binari_2.html#K5">Int_add_ovfl</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>,<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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="margrel1.html#K10">'&amp;'</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"><font color="Maroon">n</font> <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"><font color="Maroon">n</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="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="binari_2.html#D4">BINARI_2:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/> <a href="binari_2.html#K6">Int_add_udfl</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>,<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> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="margrel1.html#K10">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</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"><font color="Maroon">n</font> <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"><font color="Maroon">n</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="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="binari_2.html#D5">BINARI_2:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 

;<br/><b>then </b> <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</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="binari_4.html#T4" target="_self">Th4</a>, , , <a class="ref" href="binari_2.html#T14">BINARI_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> 0

;<br/><b>hence </b><a NAME="E30:44_2_1_1_2"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:44_2"/>
 <a href="binari_2.html#K4">Intval</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="binarith.html#K8">+</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:44"/>
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>
