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<div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds   <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> $1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1;<br/>
<a NAME="E1:19"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <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">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#T14" title="BINARITH:th.14">BINARITH:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><div class="add"><a NAME="E2:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E4:19_1_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" title="FINSEQ_2:th.97">FINSEQ_2:97</a></span>;<br/><a NAME="E5:19_1_1_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>;<br/><a NAME="E6:19_1_1_1"/>
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> 1
 <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="lab"><font color="Green" title="E19">A6</font></span>: <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="binarith.html#D1" title="BINARITH:def.1">BINARITH:def 1</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="binarith.html#T15" title="BINARITH:th.15">BINARITH:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <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="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
1

;<br/><span class="kw">hence </span><a NAME="E9:19_1_1_1"/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E5:19_1_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E7:19_1_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="binarith.html#T16" title="BINARITH:th.16">BINARITH:16</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_2"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><div class="add"><a NAME="E2:19_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <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="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E4:19_1_1_2"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" title="FINSEQ_2:th.97">FINSEQ_2:97</a></span>;<br/><a NAME="E5:19_1_1_2"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>;<br/><a NAME="E6:19_1_1_2"/>
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> 1
 <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="lab"><font color="Green" title="E19">A11</font></span>: <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="binarith.html#D1" title="BINARITH:def.1">BINARITH:def 1</a></span>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 

;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="binarith.html#T16" title="BINARITH:th.16">BINARITH:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 2</span>)</span> <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="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E9:19_1_1_2"/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E5:19_1_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E7:19_1_1_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="binarith.html#T15" title="BINARITH:th.15">BINARITH:15</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E15">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">m</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1, <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">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><span class="kw">consider </span><font color="Maroon" title="c3">t</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>, <font color="Maroon" title="c4">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <span class="kw"> such that </span><br/><a NAME="E2:19_2_1"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T117" title="FINSEQ_2:th.117">FINSEQ_2:117</a></span>;<br/><span class="lab"><font color="Green" title="E18">A15</font></span>:  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
 <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="c3">t</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c4">d</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="binari_2.html#T9" target="_self" title="BINARI_2:th.9">Th11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p2">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c4">d</font></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="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binarith.html#T20" title="BINARITH:th.20">BINARITH:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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">m</font></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> <span class="p1">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c4">d</font></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="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a></span>
;<br/><a NAME="E4:19_2_1"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p4">(<span class="default"><a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Maroon" title="c4">d</font>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="binarith.html#T20" title="BINARITH:th.20">BINARITH:20</a></span>;<br/><span class="kw">thus </span><a NAME="E5:19_2_1"/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_2_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="E1:19_2_1_2_1"/>
( <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  or <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_2_1_2_1_1"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A18</font></span>:  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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">m</font></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> <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">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E18">A15</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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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">m</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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">m</font></span>)</span></span>)</span></span>)</span> <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="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <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="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</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>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_2_1"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E1:19_2_1_2_1_1"><span class="lab"><font color="Green" title="E20">A17</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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><span class="kw">hence </span><a NAME="E4:19_2_1_2_1_1"/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1_2_1_1"><span class="lab"><font color="Green" title="E21">A18</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_2_1_2_1_2"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A20</font></span>:  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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">m</font></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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2_1"><span class="lab"><font color="Green" title="E18">A15</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="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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">m</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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">m</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_2_1"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="txt" href="#E1:19_2_1_2_1_2"><span class="lab"><font color="Green" title="E20">A19</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"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</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">m</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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">m</font></span>)</span></span>)</span></span>)</span> <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="power.html#T27" title="POWER:th.27">POWER:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p4">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</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">m</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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">m</font></span>)</span></span>)</span></span>)</span> <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="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c3">t</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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">m</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1

;<br/><span class="kw">hence </span><a NAME="E4:19_2_1_2_1_2"/>
 <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> <span class="p3">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2_1_2_1_2"><span class="lab"><font color="Green" title="E21">A20</font></span></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/></div><span class="kw">end;</span></div></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="E3:19_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:19"/>
 for <font color="Olive" title="b1">m</font> being   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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E15">A12</font></span></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/>


</div>
