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<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>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="E1:34"/><i><font color="Green" title="E28">A1</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="E2:34"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <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="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E3:34"/>
 <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="E4:34"/><b>then </b><i><font color="Green" title="E30">A3</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="E5:34"/>
 <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">y</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#E4:34"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E7:34"/>
<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="txt" href="binari_3.html#E1:34"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:34"/><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">y</font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:34"><i><font color="Green" title="E29">A2</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="E9:34"/><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/>
<b>hence </b><a NAME="E10:34"/>
 <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/>


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