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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Olive" title="b1">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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>(  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> $1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> implies <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  );<br/>
<a NAME="E1:41"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 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="nat_1.html#NM1" title="NAT_1:NM.1">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:41_1"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">and </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E5:41_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">and </span><br/><a NAME="E6:41_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">H1</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I1</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">H</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">I</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">T</font> =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">N</font> = <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1;<br/>
<a NAME="E7:41_1"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E8:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a></span>;<br/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/>
<a NAME="E10:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
;<br/>
<a NAME="E11:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E12:41_1"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E10:41_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E13:41_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E14:41_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
;<br/>
<a NAME="E15:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E16:41_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c1">h</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E10:41_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">NH</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c1">h</font>, <font color="Maroon" title="c11">NI</font> = <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/>
<a NAME="E18:41_1"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">NI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E19:41_1"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E20:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp; <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_4.html#T32" target="_self" title="BINARI_4:th.32">Th32</a></span>;<br/>
<a NAME="E21:41_1"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E22:41_1"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E23:41_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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E24:41_1"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/>
<a NAME="E25:41_1"/><span class="kw">then </span>
 <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="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E45">A18</font></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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</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> <font color="Maroon" title="c11">NI</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">NI</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:41_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E18:41_1"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E12:41_1"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="txt" href="#E13:41_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="binari_3.html#T29" title="BINARI_3:th.29">BINARI_3:29</a></span>
;<br/>
<a NAME="E27:41_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c10">NH</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E28:41_1"/>
 <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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E29:41_1"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/>
<a NAME="E30:41_1"/><span class="kw">then </span>
 <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="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="binop_2.html#K23" title="BINOP_2:func.23">+</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> <font color="Maroon" title="c10">NH</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c10">NH</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c10">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:41_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E27:41_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E11:41_1"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="txt" href="#E21:41_1"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="ref" href="binari_3.html#T29" title="BINARI_3:th.29">BINARI_3:29</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> (<span class="p5">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">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"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:41_1"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="txt" href="#E26:41_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E20:41_1"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E22:41_1"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="ref" href="binarith.html#D2" title="BINARITH:def.2">BINARITH:def 2</a></span>
<br/>.= 
<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">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"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E33:41_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:41"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:41_2"/>
1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="kw">then </span>3 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A21</font></span>: <span class="p1">(<span class="default">3 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T5" title="PEPIN:th.5">PEPIN:5</a></span>
;<br/>
<a NAME="E4:41_2"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">H</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c1">h</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">I</font> =  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c2">i</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">T</font> =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:41_2"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">and </span><br/><a NAME="E6:41_2"/><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:41_2"/><span class="lab"><font color="Green" title="E34">A25</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:41_2"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">and </span><br/><a NAME="E9:41_2"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><br/>

<a NAME="E10:41_2"/><span class="lab"><font color="Green" title="E35">A26</font></span>: 
<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#K19" title="BINOP_2:func.19">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></span>;<br/>
<a NAME="E11:41_2"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E6:41_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="binari_4.html#T9" target="_self" title="BINARI_4:th.9">Th9</a></span>;<br/>
<a NAME="E12:41_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/>
<a NAME="E13:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A27</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/>
<a NAME="E14:41_2"/>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="txt" href="#E6:41_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="binari_4.html#T9" target="_self" title="BINARI_4:th.9">Th9</a></span>;<br/>
<a NAME="E15:41_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/>
<a NAME="E16:41_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/>
<a NAME="E17:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A28</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:41_2"><span class="lab"><font color="Green" title="E35">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E18:41_2"/><span class="lab"><font color="Green" title="E38">A29</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E39">A30</font></span>: 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2 = 
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" title="POWER:th.41">POWER:41</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T29" title="PEPIN:th.29">PEPIN:29</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" title="POWER:th.41">POWER:41</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T41" title="POWER:th.41">POWER:41</a></span>
<br/>.= 
2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
2 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/>
<a NAME="E20:41_2"/><span class="lab"><font color="Green" title="E40">A31</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></span>;<br/>
<a NAME="E21:41_2"/><span class="lab"><font color="Green" title="E41">A32</font></span>: 
<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#K19" title="BINOP_2:func.19">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></span>;<br/>
<a NAME="E22:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A33</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:41_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E23:41_2"/><span class="kw">then </span>
 <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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E24:41_2"/><span class="kw">then </span>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>;<br/>
<a NAME="E25:41_2"/><span class="kw">then </span>
 <a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> (2,<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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a></span>;<br/>
<span class="kw">then </span> <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> (2,<span class="p4">(<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>)</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c1">h</font></span>)</span> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default">4 <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:41_2"><span class="lab"><font color="Green" title="E36">A27</font></span></a>, <a class="txt" href="#E21:41_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="txt" href="#E19:41_2"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c1">h</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default">2 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 3</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_2"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E18:41_2"><span class="lab"><font color="Green" title="E38">A29</font></span></a>, <a class="ref" href="binari_3.html#D1" title="BINARI_3:def.1">BINARI_3:def 1</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> (<span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c1">h</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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#K8" title="MARGREL1:func.8">TRUE</a></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> (<span class="p5">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c1">h</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> (<span class="p4">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<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> (2,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:41_2"><span class="lab"><font color="Green" title="E42">A33</font></span></a>, <a class="txt" href="#E17:41_2"><span class="lab"><font color="Green" title="E37">A28</font></span></a>, <a class="ref" href="binarith.html#D2" title="BINARITH:def.2">BINARITH:def 2</a></span>
<br/>.= 
<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> (<span class="p3">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> (2,<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> (2,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E29:41_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c1">h</font>)</span>)</span>,<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">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">i</font>)</span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:41"/>
 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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E28">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
