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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><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E24">A1</font></i>: 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E25">A2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:30"/>
<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="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:30"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="power.html#T32" title="POWER:th.32">POWER:32</a></i>;<br/>
<a NAME="E4:30"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</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>
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<a NAME="E5:30"/><b>then </b><i><font color="Green" title="E26">A3</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="nat_1.html#K1" title="NAT_1:func.1">+</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>
 
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</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
<b>by </b><i><a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>

;<br/>
<b>then </b><i><font color="Green" title="E27">A4</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <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#K3" title="SERIES_1:func.3">to_power</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
<b>by </b><i><a class="txt" href="binari_3.html#E1:30"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="binari_3.html#E5:30"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="nat_2.html#T22" title="NAT_2:th.22">NAT_2:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>
<a NAME="E8:30"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E9:30"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E10:30"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,0,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E8:30"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="binari_3.html#D1" target="_self" title="BINARI_3:def.1">Def1</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E7:30"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><br/>


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