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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">i</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 &amp;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/> <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">i</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 &amp;  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> implies  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">i</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font></span><br/>

<a NAME="E3:28"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c2">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><span class="lab"><font color="Green" title="E21">A4</font></span>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">i</font> = 
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></span>
<br/>.= 
<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">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><a NAME="E4:28_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:28_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">i</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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:28_1_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E7:28_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E8:28_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E9:28_1_1_1"/><span class="kw">then </span>
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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></span>;<br/><a NAME="E10:28_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">i</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:28_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</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="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></span>;<br/><a NAME="E12:28_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E13:28_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">then </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">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</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#D1" title="BINARI_3:def.1">BINARI_3:def 1</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E15:28_1_1_1"/>
 <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E11:28_1_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font></span><br/><div class="add"><a NAME="E2:28_1_1_2"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
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#NR2" title="XXREAL_0:NR.2">&gt;=</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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T11" title="NAT_2:th.11">NAT_2:11</a>, <a class="ref" href="power.html#T44" title="POWER:th.44">POWER:44</a></span>;<br/><a NAME="E3:28_1_1_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><a NAME="E4:28_1_1_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:28_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</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="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E6:28_1_1_2"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default"><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> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E7:28_1_1_2"/><span class="kw">then </span>
<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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">k</font> = <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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E9:28_1_1_2"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E10:28_1_1_2"/><span class="kw">then </span>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/><a NAME="E11:28_1_1_2"/><span class="kw">then </span>
 <a href="binari_4.html#K1" title="BINARI_4:func.1">MajP</a> <font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="binari_4.html#T24" target="_self" title="BINARI_4:th.24">Th24</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>:  <a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font> = 
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="binari_4.html#D2" target="_self" title="BINARI_4:def.2">Def2</a></span>
<br/>.= 
<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">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><a NAME="E13:28_1_1_2"/>
<font color="Maroon" title="c3">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="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E14:28_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">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> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_2.html#T12" title="NAT_2:th.12">NAT_2:12</a></span>;<br/><a NAME="E15:28_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/><a NAME="E16:28_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E17:28_1_1_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</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="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><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="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> = 
<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="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_2.html#T12" title="NAT_2:th.12">NAT_2:12</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>
<br/>.= 
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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>

;<br/><a NAME="E19:28_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c3">k</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E20:28_1_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">then </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">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</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#D1" title="BINARI_3:def.1">BINARI_3:def 1</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:28_1_1_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E16:28_1_1_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="nat_2.html#T22" title="NAT_2:th.22">NAT_2:22</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">then </span> <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">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">k</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="binari_2.html#D3" title="BINARI_2:def.3">BINARI_2:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_1_1_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c2">i</font>

;<br/><span class="kw">hence </span><a NAME="E23:28_1_1_2"/>
 <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:28"/>
 <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <span class="p1">(<span class="default"><a href="binari_4.html#K2" title="BINARI_4:func.2">2sComplement</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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