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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>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="nat_1.html#K1">+</a> 1 <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/><font color="Maroon">a<sub>1</sub></font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Olive">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:35"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_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="E1:35_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="series_1.html#K3">to_power</a> 1
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<a NAME="E3:35_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:35_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 = 
2 <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/>
<a NAME="E6:35_1"/>
 <a href="euclid.html#K5">0*</a> 1 <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="E7:35_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="euclid.html#K5">0*</a> 1 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="E8:35_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<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">x</font> =  <a href="euclid.html#K5">0*</a> 1 as   <a href="binarith.html#NM1">Tuple</a> of 1,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="binari_3.html#E4:35_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E10:35_1"/><i><font color="Green">E37</font></i>: 
 <a href="euclid.html#K5">0*</a> 1 <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>thus </b>1 <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="binarith.html#K4">'not'</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="binari_3.html#E1:35_1"><i><font color="Green">E32</font></i></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="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="binari_3.html#E5:35_1"><i><font color="Green">E34</font></i></a>, , , , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</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="E2:35_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:35_2"/><b>then </b><i><font color="Green">E48</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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> or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1"><b>suppose </b></a><a NAME="E1:35_2_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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/><div class="add"><a NAME="E2:35_2_1_1_1"/><b>then </b><i><font color="Green">E51</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>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:35_2_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="binarith.html#K9">add_ovfl</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>,<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</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#K7">FALSE</a> </span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K2">'xor'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <span class="p4">(<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 class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T45">BINARITH:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binari_2.html#T9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_2"><b>suppose </b></a><a NAME="E1:35_2_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:35_2_1_1_2"/><b>then </b><i><font color="Green">E54</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">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:35_2_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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">n</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="ref" href="power.html#T32">POWER:32</a></i>;<br/><a NAME="E4:35_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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">n</font></span>)</span> <a href="nat_1.html#K2">*</a> 2
 <b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E5:35_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <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">n</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">n</font></span>)</span>
 ;<br/><a NAME="E6:35_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</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">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E7:35_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <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/><a NAME="E8:35_2_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<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">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:35_2_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="binarith.html#K9">add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon">n</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">n</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">n</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>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</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="ref" href="binarith.html#T56">BINARITH:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="margrel1.html#K8">TRUE</a>  <a href="binarith.html#K2">'xor'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T45">BINARITH:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binari_2.html#T9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_3"><b>suppose </b></a><a NAME="E1:35_2_1_1_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:35_2_1_1_3"/><b>then </b><i><font color="Green">E58</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>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:35_2_1_1_3"/>
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</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="E4:35_2_1_1_3"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</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="E5:35_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</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/><b>then reconsider </b><font color="Maroon">z</font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</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="E7:35_2_1_1_3"/><i><font color="Green">E76</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
 <b>by </b><i/>;<br/><a NAME="E8:35_2_1_1_3"/><b>then </b>
<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">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:35_2_1_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="binarith.html#K9">add_ovfl</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>,<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="binari_3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></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/>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="binarith.html#K8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K2">'xor'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9">add_ovfl</a> <span class="p4">(<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 class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, , <a class="txt" href="binari_3.html#E2:35"><i><font color="Green">E38</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T45">BINARITH:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="binarith.html#K10">^</a> <span class="p2"><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></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binari_2.html#T9">BINARI_2:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <font color="Maroon">k</font></span>)</span> <a href="binarith.html#K8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:35"/>
for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(, <a class="txt" href="binari_3.html#E1:35"><i><font color="Green">E30</font></i></a>);</i><br/>


</div>
