<?xml version="1.0"?>
<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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">z</font> be   <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> ;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">z</font> <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>
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
 <a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">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#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 href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p2">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p4">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p4">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" title="BINARITH:func.9">add_ovfl</a> <span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<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>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="txt" href="binari_3.html#E2:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="binarith.html#T47" title="BINARITH:th.47">BINARITH:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="binari_2.html#T15" title="BINARI_2:th.15">BINARI_2:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p5">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="binari_3.html#T12" target="_self" title="BINARI_3:th.12">Th12</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="binari_3.html#T7" target="_self" title="BINARI_3:th.7">Th7</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="binari_3.html#T7" target="_self" title="BINARI_3:th.7">Th7</a></i>
;<br/>
<b>hence </b><a NAME="E4:24"/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K8" title="BINARITH:func.8">+</a> <span class="p1">(<span class="default"><a href="binari_2.html#K1" title="BINARI_2:func.1">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="ref" href="binari_3.html#T2" target="_self" title="BINARI_3:th.2">Th2</a></i>;<br/>


</div>
