<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">z1</font> be   <a href="finseq_2.html#NM3" title="FINSEQ_2:NM.3">Tuple</a> of 2, <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">z1</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">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <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"><a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> implies  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">z1</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">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <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"><a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:8"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T100" onmouseover="rs('finseq_2/T100')" onmouseout="rh()">FINSEQ_2:100</a></span>;<br/>
<a NAME="E4:8"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="finseq_1.html#D9" onmouseover="rs('finseq_1/D9')" onmouseout="rh()">FINSEQ_1:def 9</a></span>;<br/>
<a NAME="E5:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c1">z1</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">TRUE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" onmouseover="rs('finseq_4/T17')" onmouseout="rh()">FINSEQ_4:17</a></span>;<br/>
<a NAME="E6:8"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="finseq_4.html#T17" onmouseover="rs('finseq_4/T17')" onmouseout="rh()">FINSEQ_4:17</a></span>;<br/>
<a NAME="E7:8"/>
( 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" onmouseover="rs('finseq_1/T3')" onmouseout="rh()">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T5" onmouseover="rs('finseq_1/T5')" onmouseout="rh()">FINSEQ_1:5</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default">1 <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="binarith.html#D3" onmouseover="rs('binarith/D3')" onmouseout="rh()">BINARITH:def 3</a></span>
<br/>.= 
2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p1">(<span class="default">1 <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="#E5:8"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
;<br/>
<a NAME="E9:8"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E10:8"/><span class="kw">then </span>
1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E11:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="power.html#T24" onmouseover="rs('power/T24')" onmouseout="rh()">POWER:24</a></span>;<br/>
<a NAME="E12:8"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" onmouseover="rs('finseq_1/T3')" onmouseout="rh()">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E12">A8</font></span>: <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">IFEQ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">FALSE</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <span class="p2">(<span class="default">2 <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="binarith.html#D3" onmouseover="rs('binarith/D3')" onmouseout="rh()">BINARITH:def 3</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>
;<br/>
<a NAME="E14:8"/>
( <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</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">k1</font> &amp; <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="finseq_4.html#T17" onmouseover="rs('finseq_4/T17')" onmouseout="rh()">FINSEQ_4:17</a></span>;<br/>
<span class="kw">then </span> <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c1">z1</font> = 
<a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="binarith.html#D4" onmouseover="rs('binarith/D4')" onmouseout="rh()">BINARITH:def 4</a></span>
<br/>.= 
<a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></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"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="ref" href="finseq_1.html#D9" onmouseover="rs('finseq_1/D9')" onmouseout="rh()">FINSEQ_1:def 9</a></span>
<br/>.= 
<a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="ref" href="finsop_1.html#T5" onmouseover="rs('finsop_1/T5')" onmouseout="rh()">FINSOP_1:5</a></span>
<br/>.= 
<a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (1,<span class="p1">(<span class="default"><a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="ref" href="finsop_1.html#T11" onmouseover="rs('finsop_1/T11')" onmouseout="rh()">FINSOP_1:11</a></span>
<br/>.= 
<a href="binop_2.html#K23" title="BINOP_2:func.23">addnat</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" onmouseover="rs('finsop_1/T11')" onmouseout="rh()">FINSOP_1:11</a></span>
<br/>.= 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D23" onmouseover="rs('binop_2/D23')" onmouseout="rh()">BINOP_2:def 23</a></span>
<br/>.= 
1

;<br/>
<span class="kw">hence </span><a NAME="E16:8"/>
 <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="binari_2.html#D3" target="_self" onmouseover="rs('binari_2/D3')" onmouseout="rh()">Def3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
