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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">l</font>, <font color="Maroon" title="c3">m</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="E1:10"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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
 ;<br/>

<b>set </b><font color="Maroon" title="c4">L</font> = <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">l</font>;<br/>
<b>set </b><font color="Maroon" title="c5">M</font> = <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c3">m</font>;<br/>
<b>assume </b><a NAME="E2:10"/><i><font color="Green" title="E9">A2</font></i>: 
 <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">l</font></span>)</span>,<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="c3">m</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/>

<a NAME="E3:10"/><i><font color="Green" title="E10">A3</font></i>: 
( <font color="Maroon" title="c2">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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_4.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="binari_4.html#T8" target="_self" title="BINARI_4:th.8">Th8</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green" title="E11">A4</font></i>: 
 <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">IFEQ</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</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">l</font></span>)</span>,<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="c3">m</font></span>)</span></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<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="hidden.html#R1" title="HIDDEN:pred.1">=</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_4.html#E2:10"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>;<br/>
<i><font color="Green" title="E12">A5</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<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">l</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p3">(<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="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">IFEQ</a> <span class="p2">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <span class="p3">(<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">l</font></span>)</span>,<span class="p3">(<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="c3">m</font></span>)</span></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,0,<span class="p2">(<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></span>)</span> = 
<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">l</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"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T47" title="BINARITH:th.47">BINARITH:47</a></i>
<br/>.= 
<font color="Maroon" title="c2">l</font> <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"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E3:10"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
<br/>.= 
<font color="Maroon" title="c2">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="binari_4.html#E3:10"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/>
<a NAME="E6:10"/><i><font color="Green" title="E13">A6</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#NR4" title="XXREAL_0:NR.4">&gt;</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#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<a NAME="E7:10"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<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">l</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p3">(<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="c3">m</font></span>)</span></span>)</span></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> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E8:10"/>
contradiction
 <b>by </b><i><a class="txt" href="binari_4.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="binari_4.html#E4:10"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="binari_4.html#E5:10"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="binari_4.html#E6:10"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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