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

<b>let </b><font color="Maroon" title="c1">h</font>, <font color="Maroon" title="c2">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; ( ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) implies  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> $1,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> $1,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> );<br/>
<a NAME="E1:44"/><i><font color="Green" title="E28">A1</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" title="E29">A2</font></i>: 
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; ( ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/>

1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
0

;<br/>
<a NAME="E3:44_1"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 )
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:44_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/>
<i><font color="Green" title="E31">A4</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,0 = 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1
<b>by </b><i><a class="ref" href="binari_3.html#T26" title="BINARI_3:th.26">BINARI_3:26</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73" title="FINSEQ_2:th.73">FINSEQ_2:73</a></i>
;<br/>
<a NAME="E5:44_1"/><i><font color="Green" title="E32">A5</font></i>: 
( 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> = 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
1

;<br/>
<a NAME="E7:44_1"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
 <a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> 1,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="binari_4.html#E5:44_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="binari_4.html#D1" target="_self" title="BINARI_4:def.1">Def1</a></i>;<br/>
<a NAME="E8:44_1"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/>
<i><font color="Green" title="E35">A8</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> = 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 1
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 0</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="binari_3.html#T29" title="BINARI_3:th.29">BINARI_3:29</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="txt" href="binari_4.html#E8:44_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_5_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_5_1"/>
( ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="binari_4.html#E1:44_1"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_5_1_1"><b>suppose </b></a><a NAME="E1:44_1_5_1_1"/>
( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:44_1_5_1_1"/><b>then </b><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1 )
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="int_1.html#T21" title="INT_1:th.21">INT_1:21</a></i>;<br/><a NAME="E3:44_1_5_1_1"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:44_1_5_1_1"/>
 <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="binari_4.html#E9:44_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="binari_4.html#E2:44_1_5_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="binari_4.html#T15" target="_self" title="BINARI_4:th.15">Th15</a>, <a class="ref" href="binari_3.html#T18" title="BINARI_3:th.18">BINARI_3:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_5_1_2"><b>suppose </b></a><a NAME="E1:44_1_5_1_2"/><i><font color="Green" title="E36">A10</font></i>: 
( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:44_1_5_1_2"/><b>then </b>
<font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#T21" title="INT_1:th.21">INT_1:21</a></i>;<br/><a NAME="E3:44_1_5_1_2"/><b>then </b><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:44_1_5_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="binari_4.html#E1:44_1_5_1_2"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E5:44_1_5_1_2"/>
 <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="binari_4.html#E9:44_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="binari_4.html#E3:44_1_5_1_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="binari_4.html#T15" target="_self" title="BINARI_4:th.15">Th15</a>, <a class="ref" href="binari_3.html#T18" title="BINARI_3:th.18">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" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> 1,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/><i><font color="Green" title="E29">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:44_2"/>
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; ( ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) implies  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> )
 ;<br/>

<b>assume </b><a NAME="E2:44_2"/><i><font color="Green" title="E30">A13</font></i>: 
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp; ( ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">H</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">h</font>;<br/>
<b>set </b><font color="Maroon" title="c5">I</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">i</font>;<br/>
<b>set </b><font color="Maroon" title="c6">H1</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>;<br/>
<b>set </b><font color="Maroon" title="c7">I1</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>;<br/>
<b>set </b><font color="Maroon" title="c8">n2</font> = 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> =  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ;<br/>
<b>set </b><font color="Maroon" title="c10">T</font> =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ;<br/>
<a NAME="E3:44_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
;<br/>
<a NAME="E4:44_2"/><b>then </b><i><font color="Green" title="E31">A14</font></i>: 
(  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 &amp;  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 )
 
<b>by </b><i><a class="txt" href="binari_4.html#E2:44_2"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E6:44_2"/><i><font color="Green" title="E32">A15</font></i>: 
 <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c11">a</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="binari_4.html#T33" target="_self" title="BINARI_4:th.33">Th33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E8:44_2"/><i><font color="Green" title="E33">A16</font></i>: 
 <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c12">b</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="binari_4.html#T33" target="_self" title="BINARI_4:th.33">Th33</a></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" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 ) or ( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="binari_4.html#E2:44_2"><i><font color="Green" title="E30">A13</font></i></a></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" title="E34">A17</font></i>: 
( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c1">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E35">A18</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c13">h</font></span>)</span>
<b>by </b><i><a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c13">h</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E4:44_2_1_1_1"/><i><font color="Green" title="E36">A19</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:44_2_1_1_1"/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E6:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E38">A21</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_2_1_1_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="binari_3.html#T27" title="BINARI_3:th.27">BINARI_3:27</a></i>;<br/><a NAME="E8:44_2_1_1_1"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E39">A22</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E10:44_2_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E11:44_2_1_1_1"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E12:44_2_1_1_1"/><b>then </b>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E9:44_2_1_1_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
 <a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a></i>;<br/><a NAME="E14:44_2_1_1_1"/><i><font color="Green" title="E41">A24</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>

;<br/><a NAME="E16:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E42">A25</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">NI</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E43">A26</font></i>: <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="binari_4.html#E13:44_2_1_1_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c14">NI</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_4.html#E14:44_2_1_1_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="binari_4.html#E16:44_2_1_1_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="binari_3.html#T30" title="BINARI_3:th.30">BINARI_3:30</a></i>
;<br/><a NAME="E19:44_2_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E20:44_2_1_1_1"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E21:44_2_1_1_1"/><b>then </b><i><font color="Green" title="E44">A27</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E6:44_2_1_1_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E22:44_2_1_1_1"/><i><font color="Green" title="E45">A28</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2_1_1_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green" title="E46">A29</font></i>:  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c13">h</font>
<b>by </b><i><a class="txt" href="binari_4.html#E3:44_2_1_1_1"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="binari_4.html#E21:44_2_1_1_1"><i><font color="Green" title="E44">A27</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/><i><font color="Green" title="E47">A30</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_1"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="binari_4.html#E13:44_2_1_1_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c14">NI</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E25:44_2_1_1_1"/><i><font color="Green" title="E48">A31</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2_1_1_1"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green" title="E49">A32</font></i>:  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E18:44_2_1_1_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c14">NI</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E14:44_2_1_1_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="binari_4.html#E24:44_2_1_1_1"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon" title="c2">i</font>

;<br/><i><font color="Green" title="E50">A33</font></i>:  <a href="binari_2.html#K5" title="BINARI_2:func.5">Int_add_ovfl</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p3">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="binari_4.html#E18:44_2_1_1_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="binari_4.html#E22:44_2_1_1_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="binari_4.html#E25:44_2_1_1_1"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="binari_2.html#D4" title="BINARI_2:def.4">BINARI_2:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 

;<br/> <a href="binari_2.html#K6" title="BINARI_2:func.6">Int_add_udfl</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="binari_4.html#E18:44_2_1_1_1"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="binari_4.html#E22:44_2_1_1_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="binari_4.html#E25:44_2_1_1_1"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="binari_2.html#D5" title="BINARI_2:def.5">BINARI_2:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 

;<br/><b>then </b> <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E6:44_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="binari_4.html#E8:44_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="binari_4.html#E23:44_2_1_1_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="binari_4.html#E26:44_2_1_1_1"><i><font color="Green" title="E49">A32</font></i></a>, <a class="txt" href="binari_4.html#E27:44_2_1_1_1"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="binari_2.html#T14" title="BINARI_2:th.14">BINARI_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0

;<br/><b>hence </b><a NAME="E30:44_2_1_1_1"/>
 <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">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" title="E34">A34</font></i>: 
( <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c2">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E35">A35</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c13">i</font></span>)</span>
<b>by </b><i><a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c13">i</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E4:44_2_1_1_2"/><i><font color="Green" title="E36">A36</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:44_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E6:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E37">A37</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A38</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_4.html#E3:44_2_1_1_2"><i><font color="Green" title="E35">A35</font></i></a>, <a class="ref" href="binari_3.html#T27" title="BINARI_3:th.27">BINARI_3:27</a></i>;<br/><a NAME="E8:44_2_1_1_2"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">h</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E9:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E39">A39</font></i>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E10:44_2_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E11:44_2_1_1_2"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E12:44_2_1_1_2"/><b>then </b>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E9:44_2_1_1_2"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E40">A40</font></i>: 
 <a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a></i>;<br/><a NAME="E14:44_2_1_1_2"/><i><font color="Green" title="E41">A41</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>

;<br/><a NAME="E16:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E42">A42</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">h</font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">NH</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><i><font color="Green" title="E43">A43</font></i>: <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="txt" href="binari_4.html#E13:44_2_1_1_2"><i><font color="Green" title="E40">A40</font></i></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c14">NH</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_4.html#E14:44_2_1_1_2"><i><font color="Green" title="E41">A41</font></i></a>, <a class="txt" href="binari_4.html#E16:44_2_1_1_2"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="binari_3.html#T30" title="BINARI_3:th.30">BINARI_3:30</a></i>
;<br/><a NAME="E19:44_2_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E20:44_2_1_1_2"/><b>then </b>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></i>;<br/><a NAME="E21:44_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">A44</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E6:44_2_1_1_2"><i><font color="Green" title="E37">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E22:44_2_1_1_2"/><i><font color="Green" title="E45">A45</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2_1_1_2"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green" title="E46">A46</font></i>:  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_2"><i><font color="Green" title="E38">A38</font></i></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c13">i</font>
<b>by </b><i><a class="txt" href="binari_4.html#E3:44_2_1_1_2"><i><font color="Green" title="E35">A35</font></i></a>, <a class="txt" href="binari_4.html#E21:44_2_1_1_2"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/><i><font color="Green" title="E47">A47</font></i>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E1:44_2_1_1_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="txt" href="binari_4.html#E13:44_2_1_1_2"><i><font color="Green" title="E40">A40</font></i></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c14">NH</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/><a NAME="E25:44_2_1_1_2"/><i><font color="Green" title="E48">A48</font></i>: 
<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:44_2_1_1_2"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then </b><i><font color="Green" title="E49">A49</font></i>:  <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E18:44_2_1_1_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c14">NH</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E14:44_2_1_1_2"><i><font color="Green" title="E41">A41</font></i></a>, <a class="txt" href="binari_4.html#E24:44_2_1_1_2"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon" title="c1">h</font>

;<br/><i><font color="Green" title="E50">A50</font></i>:  <a href="binari_2.html#K5" title="BINARI_2:func.5">Int_add_ovfl</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p3">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_2"><i><font color="Green" title="E38">A38</font></i></a>, <a class="txt" href="binari_4.html#E18:44_2_1_1_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="binari_4.html#E22:44_2_1_1_2"><i><font color="Green" title="E45">A45</font></i></a>, <a class="txt" href="binari_4.html#E25:44_2_1_1_2"><i><font color="Green" title="E48">A48</font></i></a>, <a class="ref" href="binari_2.html#D4" title="BINARI_2:def.4">BINARI_2:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 

;<br/> <a href="binari_2.html#K6" title="BINARI_2:func.6">Int_add_udfl</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E7:44_2_1_1_2"><i><font color="Green" title="E38">A38</font></i></a>, <a class="txt" href="binari_4.html#E18:44_2_1_1_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="binari_4.html#E22:44_2_1_1_2"><i><font color="Green" title="E45">A45</font></i></a>, <a class="txt" href="binari_4.html#E25:44_2_1_1_2"><i><font color="Green" title="E48">A48</font></i></a>, <a class="ref" href="binari_2.html#D5" title="BINARI_2:def.5">BINARI_2:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 

;<br/><b>then </b> <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E6:44_2"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="binari_4.html#E8:44_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="binari_4.html#E23:44_2_1_1_2"><i><font color="Green" title="E46">A46</font></i></a>, <a class="txt" href="binari_4.html#E26:44_2_1_1_2"><i><font color="Green" title="E49">A49</font></i></a>, <a class="txt" href="binari_4.html#E27:44_2_1_1_2"><i><font color="Green" title="E50">A50</font></i></a>, <a class="ref" href="binari_2.html#T14" title="BINARI_2:th.14">BINARI_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0

;<br/><b>hence </b><a NAME="E30:44_2_1_1_2"/>
 <a href="binari_2.html#K4" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">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" title="BINARI_2:func.4">Intval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:44"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="binari_4.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="binari_4.html#E2:44"><i><font color="Green" title="E29">A12</font></i></a>);</i><br/>


</div>
