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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">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> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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="Olive" title="b1">z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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></span><br/>

<a NAME="E2:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> (<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T23" target="_self" title="BINARI_3:th.23">Th24</a></span>;<br/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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#K6" title="BINARITH:func.6">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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#K4" title="SERIES_1:func.4">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#K4" title="SERIES_1:func.4">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#K6" title="BINARITH:func.6">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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#K8" title="BINARITH:func.8">add_ovfl</a> (<span class="p4">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'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#K4" title="SERIES_1:func.4">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#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p3">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'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#K6" title="BINARITH:func.6">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#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#T21" title="BINARITH:th.21">BINARITH:21</a></span>
<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#K6" title="BINARITH:func.6">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#K4" title="SERIES_1:func.4">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#K6" title="BINARITH:func.6">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#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_2.html#T13" title="BINARI_2:th.13">BINARI_2:13</a></span>
<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#K6" title="BINARITH:func.6">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#K4" title="SERIES_1:func.4">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#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T11" target="_self" title="BINARI_3:th.11">Th12</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="binari_3.html#T6" target="_self" title="BINARI_3:th.6">Th7</a></span>
<br/>.= 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="binari_3.html#T6" target="_self" title="BINARI_3:th.6">Th7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:24"/>
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T2" target="_self" title="BINARI_3:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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