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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 ;<br/>

<b>thus </b><a NAME="E3:32"/>
( <font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font> implies <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">y</font> being  <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">a<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>   st <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Olive">y</font> holds <br/><font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1;<br/>
<a NAME="E2:32_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> 1
 ;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><a NAME="E2:32_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> 1
 ;<br/>

<a NAME="E3:32_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>;<br/>
<b>assume </b><a NAME="E4:32_1_1"/><i><font color="Green">E38</font></i>: 
1 <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:32_1_1"/><i><font color="Green">E39</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1 <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:32_1_1"/><i><font color="Green">E46</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E7:32_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 
<b>by </b><i><a class="txt" href="binari_3.html#E1:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<i><font color="Green">E50</font></i>: 1 = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="txt" href="binari_3.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="binari_3.html#E1:32_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span>,0,<a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E2:32_1"><i><font color="Green">E34</font></i></a>, </i>
;<br/>
<a NAME="E9:32_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:32_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/>
<a NAME="E3:32_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E4:32_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T80">NAT_1:80</a></i>;<br/>
<a NAME="E5:32_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E6:32_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_3.html#E1:32_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon">k</font> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:32_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:32_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">y</font> be   <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:32_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E4:32_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:32_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:32_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E7:32_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E8:32_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E9:32_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:32_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="txt" href="binari_3.html#E1:32_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">y</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E2:32_1"><i><font color="Green">E34</font></i></a>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 

;<br/>
<a NAME="E12:32_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:32_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:32"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E14:32_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E15:32_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E16:32_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span>
 
;<br/>
<a NAME="E17:32_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E18:32_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="binari_3.html#T2" target="_self">Th2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E19:32_1_2"/>
 <a href="euclid.html#K5">0*</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <a href="margrel1.html#K6">BOOLEAN</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E20:32_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">m</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E21:32_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">m</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">m</font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E23:32_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">0</span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="binari_3.html#E1:32_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/>
<a NAME="E24:32_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">y1</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:32_1"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="binari_2.html#T11">BINARI_2:11</a></i>;<br/>
<a NAME="E25:32_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<a NAME="E26:32_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E27:32_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binari_3.html#T2" target="_self">Th2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:32_1"/>
for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, <a class="txt" href="binari_3.html#E1:32"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:32_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binari_3.html#T1" target="_self">Th1</a>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:32"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:32_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E3:32_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:32_2"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E5:32_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E6:32_2"/><i><font color="Green">E108</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:32_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, </i>;<br/><a NAME="E8:32_2"/>
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E9:32_2"/><b>then </b><i><font color="Green">E110</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:32_2"/><i><font color="Green">E111</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:32_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:32_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:32_2"/><b>then </b><i><font color="Green">E112</font></i>: 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E14:32_2"/><b>then </b><i><font color="Green">E113</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_2.html#T13">NAT_2:13</a></i>;<br/><a NAME="E15:32_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E16:32_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E17:32_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_2.html#T19">NAT_2:19</a></i>;<br/><a NAME="E18:32_2"/><i><font color="Green">E115</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E19:32_2"/><b>then </b><i><font color="Green">E116</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:32_2"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/><a NAME="E21:32_2"/><b>then </b><i><font color="Green">E117</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, , , <a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="nat_2.html#T18">NAT_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><a NAME="E23:32_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="binari_3.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_2.html#T20">NAT_2:20</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span>,0,<a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="binari_3.html#E2:32"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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