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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> implies  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E1:37"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:37_1"/>
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="binari_3.html#T26" target="_self" title="BINARI_3:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1"/>
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="binari_3.html#T7" target="_self" title="BINARI_3:th.7">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:37"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> implies  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 ;<br/>

<b>assume </b><a NAME="E2:37_2"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:37_2"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:37_2"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T5" target="_self" title="BINARI_3:th.5">Th5</a></i>;<br/>
<a NAME="E5:37_2"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E6:37_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">0n</font> =  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="binari_3.html#E5:37_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E8:37_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:37_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:37_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c3">0n</font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E3:37_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="binari_3.html#E33"><i><font color="Green" title="E25">Lm4</font></i></a></i>;<br/>
<a NAME="E10:37_2"/><b>then </b>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="binari_3.html#T24" target="_self" title="BINARI_3:th.24">Th24</a></i>;<br/>
<a NAME="E11:37_2"/><b>then </b>
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/>
<a NAME="E12:37_2"/><b>then </b><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font>, <a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font> <a href="binarith.html#R1" title="BINARITH:pred.1">are_summable</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#D10" title="BINARITH:def.10">BINARITH:def 10</a></i>;<br/>
<b>thus </b> <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E2:37_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="binari_3.html#T34" target="_self" title="BINARI_3:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E12:37_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="binarith.html#T48" title="BINARITH:th.48">BINARITH:48</a></i>
<br/>.= 
<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="binari_3.html#E1:37_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="binari_3.html#E2:37_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="binari_3.html#T12" target="_self" title="BINARI_3:th.12">Th12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:37"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="binari_3.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="binari_3.html#E2:37"><i><font color="Green" title="E31">A2</font></i></a>);</i><br/>


</div>
