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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>] <span class="kw">means </span>( $4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> ($2,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>) &amp; $5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> ($3,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>) );<br/>
<a NAME="E1:84_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">z</font>, <font color="Olive" title="b3">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>  ex <font color="Olive" title="b4">z1</font>, <font color="Olive" title="b5">w1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>,<font color="Olive" title="b3">w</font>,<font color="Olive" title="b4">z1</font>,<font color="Olive" title="b5">w1</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>  ex <font color="Olive" title="b3">z1</font>, <font color="Olive" title="b4">w1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>,<font color="Olive" title="b3">z1</font>,<font color="Olive" title="b4">w1</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">w</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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">  ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">w1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font>,<font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">w1</font>]</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">z1</font> =  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c3">z</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>) as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#T14" target="_self" onmouseover="rs('descip_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w1</font> =  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<font color="Maroon" title="c4">w</font>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>) as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#T14" target="_self" onmouseover="rs('descip_1/T14')" onmouseout="rh()">Th14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">z1</font>
; <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">  ex <font color="Olive" title="b1">w1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">z1</font>,<font color="Olive" title="b1">w1</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">w1</font>
; <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">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">w1</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:84_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c6">w1</font>]
 ; <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><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c2">C0</font> =  <a href="descip_1.html#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#T1" target="_self" onmouseover="rs('descip_1/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E3:84_1_1"/>
(  <a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 56 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> &amp; 28 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 56 &amp; 28 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 56 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 28 )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">D0</font> =  <a href="descip_1.html#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="descip_1.html#T2" target="_self" onmouseover="rs('descip_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">D</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:84_1_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
( <font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C0</font> &amp; <font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D0</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_2.html#S3" onmouseover="rs('recdef_2/S3')" onmouseout="rh()">RECDEF_2:sch 3</a>(<a class="txt" href="#E1:84_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> $1</span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> $1</span>)</span></span>)</span>;<br/>
<a NAME="E7:84_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 16 holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 48 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">OUT</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 48 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><span class="kw"> such that </span><br/><a NAME="E9:84_1_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">OUT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 16 &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 16 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c6">OUT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E7:84_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>);<br/></span>
<a NAME="E10:84_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">OUT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 16
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:84_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">OUT</font> = <font color="Maroon" title="c6">OUT</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 16 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default">48 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" onmouseover="rs('finseq_2/T92')" onmouseout="rh()">FINSEQ_2:92</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E41">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="84_1_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">i</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>   st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 holds <br/>( <font color="Maroon" title="c7">OUT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>) &amp; <font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be    <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> ; <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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 implies ( <font color="Maroon" title="c7">OUT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) &amp; <font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:84_1_1_2"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 )
 ; <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="c7">OUT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) &amp; <font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) )</span><br/><a NAME="E2:84_1_1_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:84_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E4:84_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> 16
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:84_1_1_2"/>
( <font color="Maroon" title="c7">OUT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) &amp; <font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E9:84_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></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><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E13:84_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 16 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default">48 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> ex <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">D</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default">28 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> st <br/>( <font color="Olive" title="b2">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K2" title="DESCIP_1:func.2">Op-Left</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; <font color="Olive" title="b3">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K3" title="DESCIP_1:func.3">Op-Right</a> (<span class="p1">(<span class="default"><a href="descip_1.html#K35" title="DESCIP_1:func.35">DES-PC1</a> <font color="Maroon" title="c1">Key</font></span>)</span>,28) &amp; (  for <font color="Olive" title="b4">i</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>   st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 15 holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K36" title="DESCIP_1:func.36">DES-PC2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="descip_1.html#K9" title="DESCIP_1:func.9">^</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">C</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span>) &amp; <font color="Olive" title="b3">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="descip_1.html#K8" title="DESCIP_1:func.8">Op-Shift</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">D</font> <a href="descip_1.html#K13" title="DESCIP_1:func.13">.</a> <font color="Olive" title="b4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="descip_1.html#K37" title="DESCIP_1:func.37">bitshift_DES</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span>) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:84_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E6:84_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></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>
