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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="armstrng.html#NM5" title="ARMSTRNG:NM.5">Full-family</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>consider </b><font color="Maroon" title="c3">G</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:98"/>
<font color="Maroon" title="c3">G</font> <a href="armstrng.html#R5" title="ARMSTRNG:pred.5">is_generator-set_of</a>  <a href="armstrng.html#K9" title="ARMSTRNG:func.9">saturated-subsets</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E3:98"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">X</font> <a href="armstrng.html#K10" title="ARMSTRNG:func.10">deps_encl_by</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="armstrng.html#T45" target="_self" title="ARMSTRNG:th.45">Th45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">H</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E5:98"/><i><font color="Green" title="E56">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E6:98"/>
<font color="Maroon" title="c4">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">H</font> = <font color="Maroon" title="c4">H</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">G</font> <b>by </b><i><a class="txt" href="armstrng.html#E5:98"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V12</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E9:98"/><i><font color="Green" title="E57">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E58">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>thus </b><a NAME="E1:98_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:98_1"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 ;<br/><a NAME="E3:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E5:98_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="armstrng.html#E2:98_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:98"/><b>then </b><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2"/>
(  not <font color="Maroon" title="c3">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> or <font color="Maroon" title="c3">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_1"><b>suppose </b></a><a NAME="E1:98_2_1"/><i><font color="Green" title="E60">A8</font></i>: 
not <font color="Maroon" title="c3">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c7">R</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><a NAME="E2:98_2_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:98_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:98_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:98"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">R</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font></span>)</span> ;<br/><b>set </b><font color="Maroon" title="c9">BD</font> =  <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #);<br/><b>take </b>
 <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
;<br/><b>thus </b><a NAME="E4:98_2_1"/>
( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  the <a href="armstrng.html#U2" title="ARMSTRNG:sel.2">Domains</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c10">Ds</font> =  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #);<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c13">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:98_2_1_2_1"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:98_2_1_2_1"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c13">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">A</font> = <font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c15">B</font> = <font color="Maroon" title="c13">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #) ;<br/><a NAME="E6:98_2_1_2_1"/>
<font color="Maroon" title="c14">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c15">B</font>, <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">f</font>, <font color="Maroon" title="c17">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">ARMSTRNG:def 8</a><br/>

<a NAME="E1:98_2_1_2_1_1"/>
( <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:98_2_1_2_1_1"/>
( <font color="Maroon" title="c16">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font> implies <font color="Maroon" title="c16">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:98_2_1_2_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_1_2_1"><i><font color="Green" title="E61">A9</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:98_2_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
 ;<br/><b>then consider </b><font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c13">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)<b> such that </b><br/><a NAME="E4:98_2_1_2"/><i><font color="Green" title="E61">A10</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:98_2_1_2"/>
<font color="Maroon" title="c12">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c13">B</font>, <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
 ;<br/><a NAME="E6:98_2_1_2"/>
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c12">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c13">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_1"><i><font color="Green" title="E60">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E7:98_2_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="armstrng.html#E4:98_2_1_2"><i><font color="Green" title="E61">A10</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:98_2_1"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c8">R</font> #)
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2"><b>suppose </b></a><a NAME="E1:98_2_2"/><i><font color="Green" title="E60">A11</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">H</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="armstrng.html#E5:98"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E3:98_2_2"/><i><font color="Green" title="E61">A12</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c7">n</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> ] <b>means </b>for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/>( ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> implies $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> implies $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) );<br/><div><i><font color="Green" title="E62">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) );<br/><a NAME="E1:98_2_2_1"/><i><font color="Green" title="E63">A14</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/> ex <font color="Olive" title="b2">x</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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:98_2_2_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_1_1_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> or  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_2_1_1_1_1"/><i><font color="Green" title="E64">A15</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">b</font> =  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  as    <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>  ;<br/><b>take </b>
<font color="Maroon" title="c10">b</font>
;<br/><b>thus </b><a NAME="E3:98_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">b</font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_1_1_1_1"><i><font color="Green" title="E64">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_2_1_1_1_2"/><i><font color="Green" title="E64">A16</font></i>: 
not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c10">b</font> =  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  as    <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>  ;<br/><b>take </b>
<font color="Maroon" title="c10">b</font>
;<br/><b>thus </b><a NAME="E3:98_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">b</font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_1_1_1_2"><i><font color="Green" title="E64">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <b> such that </b><br/><a NAME="E3:98_2_2_1"/><i><font color="Green" title="E64">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>and </b><br/><a NAME="E4:98_2_2_1"/><i><font color="Green" title="E65">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="armstrng.html#E1:98_2_2_1"><i><font color="Green" title="E63">A14</font></i></a>);<br/></i><a NAME="E5:98_2_2_1"/><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:98_2_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:98_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_1"><i><font color="Green" title="E66">A19</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c7">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/><b>take </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font>;<br/><b>thus </b><a NAME="E9:98_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">y</font>]
 <b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_1"><i><font color="Green" title="E65">A18</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c8">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</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> <b> such that </b><br/><a NAME="E6:98_2_2"/><i><font color="Green" title="E63">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="armstrng.html#E4:98_2_2"><i><font color="Green" title="E62">A13</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c9">R</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>take </b><font color="Maroon" title="c11">i</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><i><font color="Green" title="E64">A21</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> = 
<font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="ref" href="armstrng.html#T5" target="_self" title="ARMSTRNG:th.5">Th5</a></i>
<br/>.= 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="binari_3.html#T26" title="BINARI_3:th.26">BINARI_3:26</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_2"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="binari_3.html#T7" title="BINARI_3:th.7">BINARI_3:7</a></i>
;<br/></div><b>end;</b></div><a NAME="E8:98_2_2"/><b>then </b><i><font color="Green" title="E64">A22</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E11:98"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/><a NAME="E9:98_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_2_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:98_2_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> st <br/>( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:98_2_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">R</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> :  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font></span>)</span> <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2"><i><font color="Green" title="E64">A22</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">BD</font> =  <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #);<br/><b>take </b>
 <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
;<br/><b>thus </b><a NAME="E11:98_2_2"/>
( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  the <a href="armstrng.html#U2" title="ARMSTRNG:sel.2">Domains</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c12">Ds</font> =  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #);<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_2_2_4_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">A</font>, <font color="Maroon" title="c15">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:98_2_2_4_1"/><i><font color="Green" title="E65">A23</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E4:98_2_2_4_1"/><i><font color="Green" title="E66">A24</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">A'</font> = <font color="Maroon" title="c14">A</font>, <font color="Maroon" title="c17">B'</font> = <font color="Maroon" title="c15">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #) ;<br/><a NAME="E6:98_2_2_4_1"/>
<font color="Maroon" title="c16">A'</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c17">B'</font>, <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">f</font>, <font color="Maroon" title="c19">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="armstrng.html#U3" title="ARMSTRNG:sel.3">Relationship</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">ARMSTRNG:def 8</a><br/>

<b>assume </b><a NAME="E1:98_2_2_4_1_1"/><i><font color="Green" title="E67">A25</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">A'</font>
 ;<br/>

<a NAME="E2:98_2_2_4_1_1"/>
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c20">Rf</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font><b> such that </b><br/><a NAME="E4:98_2_2_4_1_1"/><i><font color="Green" title="E68">A26</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Rf</font>
 <b>and </b><br/><a NAME="E5:98_2_2_4_1_1"/><i><font color="Green" title="E69">A27</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c20">Rf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span>
 ;<br/>
<a NAME="E6:98_2_2_4_1_1"/>
<font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c21">Rg</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font><b> such that </b><br/><a NAME="E8:98_2_2_4_1_1"/><i><font color="Green" title="E70">A28</font></i>: 
<font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Rg</font>
 <b>and </b><br/><a NAME="E9:98_2_2_4_1_1"/><i><font color="Green" title="E71">A29</font></i>: 
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c21">Rg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon" title="c22">fi</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:98_2_2_4_1_1"/><i><font color="Green" title="E72">A30</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c20">Rf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_1_1"><i><font color="Green" title="E69">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">gi</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:98_2_2_4_1_1"/><i><font color="Green" title="E73">A31</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c21">Rg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98_2_2_4_1_1"><i><font color="Green" title="E71">A29</font></i></a></i>;<br/>
<i><font color="Green" title="E74">A32</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">g</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green" title="E70">A28</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f</font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green" title="E68">A26</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_1_1_2"/><i><font color="Green" title="E75">A33</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_1_1"><i><font color="Green" title="E74">A32</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon" title="c24">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_1_1_2"/><i><font color="Green" title="E76">A34</font></i>: 
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_1_1_2"/><i><font color="Green" title="E77">A35</font></i>: 
<font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2"><i><font color="Green" title="E75">A33</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2"><i><font color="Green" title="E76">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2"><i><font color="Green" title="E75">A33</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2"><i><font color="Green" title="E76">A34</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c26">nbf</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font>;<br/><b>set </b><font color="Maroon" title="c27">nbg</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font>;<br/><b>set </b><font color="Maroon" title="c28">nf</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c29">ng</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span>;<br/><a NAME="E5:98_2_2_4_1_1_2"/><i><font color="Green" title="E78">A36</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E6:98_2_2_4_1_1_2"/><i><font color="Green" title="E79">A37</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">Mx</font> = <font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c7">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_1_1_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c31">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_1_1_2_1"/><i><font color="Green" title="E80">A38</font></i>: 
<font color="Maroon" title="c31">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_4_1_1_2_1_1"/>
( <font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> or  not <font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:98_2_2_4_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E81">A39</font></i>: 
<font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E4:98_2_2_4_1"><i><font color="Green" title="E66">A24</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><i><font color="Green" title="E82">A40</font></i>: <font color="Maroon" title="c30">Mx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font> = 
<font color="Maroon" title="c30">Mx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_1_1_2"><i><font color="Green" title="E78">A36</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2"><i><font color="Green" title="E77">A35</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_1"><i><font color="Green" title="E81">A39</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">Mx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">Mx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2_1_1_1"><i><font color="Green" title="E82">A40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:98_2_2_4_1_1_2_1_1_2"/><i><font color="Green" title="E81">A41</font></i>: 
not <font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:98_2_2_4_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c32">xx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E82">A42</font></i>: 
<font color="Maroon" title="c32">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>and </b><br/><a NAME="E3:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E83">A43</font></i>: 
not <font color="Maroon" title="c32">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1_2_1_1_2"><i><font color="Green" title="E81">A41</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">xx</font> = <font color="Maroon" title="c32">xx</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E82">A42</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">Mxx</font> = <font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c7">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/>
<a NAME="E6:98_2_2_4_1_1_2_1_1_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E84">A44</font></i>: <font color="Maroon" title="c34">Mxx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font> = 
<font color="Maroon" title="c34">Mxx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E83">A43</font></i></a></i>
;<br/>
<i><font color="Green" title="E85">A45</font></i>: <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xx</font>
<b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_1_1"><i><font color="Green" title="E67">A25</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E82">A42</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xx</font>
<b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E82">A42</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E9:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E86">A46</font></i>: 
<font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green" title="E68">A26</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1"><i><font color="Green" title="E72">A30</font></i></a></i>;<br/>
<a NAME="E10:98_2_2_4_1_1_2_1_1_2_1"/><i><font color="Green" title="E87">A47</font></i>: 
<font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green" title="E70">A28</font></i></a>, <a class="txt" href="armstrng.html#E13:98_2_2_4_1_1"><i><font color="Green" title="E73">A31</font></i></a></i>;<br/>
<i><font color="Green" title="E88">A48</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Mxx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E84">A44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">xx</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E85">A45</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E86">A46</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E87">A47</font></i></a>, <a class="ref" href="binari_3.html#T2" title="BINARI_3:th.2">BINARI_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Mxx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E84">A44</font></i></a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">Mx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_1_1_2"><i><font color="Green" title="E79">A37</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2_1"><i><font color="Green" title="E80">A38</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1_2_1_1_2_1"><i><font color="Green" title="E88">A48</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:98_2_2_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c23">gi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>then </b><font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c22">fi</font></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_1_1"><i><font color="Green" title="E70">A28</font></i></a>, <a class="txt" href="armstrng.html#E13:98_2_2_4_1_1"><i><font color="Green" title="E73">A31</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:98_2_2_4_1_1"><i><font color="Green" title="E68">A26</font></i></a>, <a class="txt" href="armstrng.html#E11:98_2_2_4_1_1"><i><font color="Green" title="E72">A30</font></i></a></i>
;<br/><b>hence </b><a NAME="E11:98_2_2_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_1_1_2"><i><font color="Green" title="E76">A34</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:98_2_2_4_1_1"/>
<font color="Maroon" title="c18">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">B'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:98_2_2_4_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_1"><i><font color="Green" title="E65">A23</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:98_2_2_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
 ;<br/><b>then consider </b><font color="Maroon" title="c14">A</font>, <font color="Maroon" title="c15">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)<b> such that </b><br/><a NAME="E4:98_2_2_4"/><i><font color="Green" title="E65">A49</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c15">B</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:98_2_2_4"/><i><font color="Green" title="E66">A50</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c15">B</font>, <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">gg</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:98_2_2_4_2"/><i><font color="Green" title="E67">A51</font></i>: 
<font color="Maroon" title="c16">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E2:98_2_2_4_2"/><i><font color="Green" title="E68">A52</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">gg</font>
 <b>and </b><br/><a NAME="E3:98_2_2_4_2"/><i><font color="Green" title="E69">A53</font></i>: 
not <font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">gg</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c17">gg</font> = <font color="Maroon" title="c16">gg</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">G</font> <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_2"><i><font color="Green" title="E67">A51</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c18">bx</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:98_2_2_4_2"/><i><font color="Green" title="E70">A54</font></i>: 
( <font color="Maroon" title="c18">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">B</font> &amp; not <font color="Maroon" title="c18">bx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">gg</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2"><i><font color="Green" title="E69">A53</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">bx</font> = <font color="Maroon" title="c18">bx</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green" title="E70">A54</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:98_2_2_4_2"/><i><font color="Green" title="E71">A55</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">H</font>
 <b>and </b><br/><a NAME="E10:98_2_2_4_2"/><i><font color="Green" title="E72">A56</font></i>: 
<font color="Maroon" title="c5">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">gg</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2"><i><font color="Green" title="E60">A11</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E11:98_2_2_4_2"/><i><font color="Green" title="E73">A57</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:98_2_2_4_2"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:98_2_2_4_2"/><i><font color="Green" title="E74">A58</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">k</font> = <font color="Maroon" title="c21">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E16:98_2_2_4_2"/><i><font color="Green" title="E75">A59</font></i>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:98_2_2_4_2"><i><font color="Green" title="E73">A57</font></i></a>, <a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c23">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:98_2_2_4_2"/><i><font color="Green" title="E76">A60</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c23">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c24">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:98_2_2_4_2"/><i><font color="Green" title="E77">A61</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c24">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E21:98_2_2_4_2"/><i><font color="Green" title="E78">A62</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:98_2_2_4_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">x'</font> = <font color="Maroon" title="c25">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:98_2_2_4_2_1"/>
<font color="Maroon" title="c23">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E18:98_2_2_4_2"><i><font color="Green" title="E76">A60</font></i></a></i>;<br/><a NAME="E4:98_2_2_4_2_1"/><b>then </b>
<font color="Maroon" title="c23">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E5:98_2_2_4_2_1"/><b>then </b>
<font color="Maroon" title="c23">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_2_2_4_2_1"/>
<font color="Maroon" title="c23">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c25">f</font> = <font color="Maroon" title="c23">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="armstrng.html#E21:98_2_2_4_2"><i><font color="Green" title="E78">A62</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E24:98_2_2_4_2"/><i><font color="Green" title="E79">A63</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E18:98_2_2_4_2"><i><font color="Green" title="E76">A60</font></i></a></i>;<br/><a NAME="E25:98_2_2_4_2"/><i><font color="Green" title="E80">A64</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E9:98"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:98_2_2_4_2_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">D</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c27">x'</font> = <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:98_2_2_4_2_2"/>
<font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green" title="E77">A61</font></i></a></i>;<br/><a NAME="E4:98_2_2_4_2_2"/><b>then </b>
<font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E5:98_2_2_4_2_2"/><b>then </b>
<font color="Maroon" title="c24">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a></i>;<br/><b>hence </b><a NAME="E6:98_2_2_4_2_2"/>
<font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c26">g</font> = <font color="Maroon" title="c24">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="txt" href="armstrng.html#E25:98_2_2_4_2"><i><font color="Green" title="E80">A64</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E28:98_2_2_4_2"/><i><font color="Green" title="E81">A65</font></i>: 
<font color="Maroon" title="c26">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green" title="E77">A61</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:98_2_2_4_2_3"/><i><font color="Green" title="E82">A66</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E21:98_2_2_4_2"><i><font color="Green" title="E78">A62</font></i></a>, <a class="txt" href="armstrng.html#E25:98_2_2_4_2"><i><font color="Green" title="E80">A64</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_2_3"/><i><font color="Green" title="E83">A67</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_2_3"/><b>then </b><i><font color="Green" title="E84">A68</font></i>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_2_3"><i><font color="Green" title="E82">A66</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">a</font> = <font color="Maroon" title="c27">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E1:98_2_2_4_2_3"><i><font color="Green" title="E82">A66</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_2_3"><i><font color="Green" title="E83">A67</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c29">bs0</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>set </b><font color="Maroon" title="c30">bsi</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span>;<br/><i><font color="Green" title="E85">A69</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> = 
<font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3"><i><font color="Green" title="E84">A68</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E18:98_2_2_4_2"><i><font color="Green" title="E76">A60</font></i></a></i>
;<br/><a NAME="E6:98_2_2_4_2_3"/><i><font color="Green" title="E86">A70</font></i>: 
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green" title="E77">A61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">Ma</font> = <font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c7">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/><b>set </b><font color="Maroon" title="c32">L</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c33">R</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2_4_2_3_2"><b>now </b></a><div class="add"><b>thus </b> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>
;<br/><b>let </b><font color="Maroon" title="c34">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:98_2_2_4_2_3_2"/><i><font color="Green" title="E87">A71</font></i>: 
<font color="Maroon" title="c34">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:98_2_2_4_2_3_2"/><b>then </b><i><font color="Green" title="E88">A72</font></i>: 
<font color="Maroon" title="c34">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c35">nj</font> = <font color="Maroon" title="c34">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="armstrng.html#E2:98_2_2_4_2_3_2"><i><font color="Green" title="E87">A71</font></i></a></i>;<br/><i><font color="Green" title="E89">A73</font></i>: <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="binari_3.html#T26" title="BINARI_3:th.26">BINARI_3:26</a></i>
<br/>.= 
<font color="Maroon" title="c7">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="euclid.html#D4" title="EUCLID:def.4">EUCLID:def 4</a></i>
;<br/><a NAME="E6:98_2_2_4_2_3_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E90">A74</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">nj</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="txt" href="armstrng.html#E5:98_2_2_4_2_3_2"><i><font color="Green" title="E89">A73</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green" title="E91">A75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Ma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E7:98_2_2_4_2_3_2"><i><font color="Green" title="E90">A74</font></i></a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_4_2_3_2_5"/>
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">nj</font> or <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">nj</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5_1"><b>suppose </b></a><a NAME="E1:98_2_2_4_2_3_2_5_1"/><i><font color="Green" title="E92">A76</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">nj</font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_2_3_2_5_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E93">A77</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">nj</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="txt" href="armstrng.html#E16:98_2_2_4_2"><i><font color="Green" title="E75">A59</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_1"><i><font color="Green" title="E92">A76</font></i></a>, <a class="ref" href="armstrng.html#T8" target="_self" title="ARMSTRNG:th.8">Th8</a></i>
;<br/><a NAME="E4:98_2_2_4_2_3_2_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Ma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E5:98_2_2_4_2_3_2_5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_2_3_2"><i><font color="Green" title="E91">A75</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2_5_1"><i><font color="Green" title="E93">A77</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_4_2_3_2_5_2"><b>suppose </b></a><a NAME="E1:98_2_2_4_2_3_2_5_2"/><i><font color="Green" title="E92">A78</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">nj</font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_4_2_3_2_5_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c31">Ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E93">A79</font></i>: <font color="Maroon" title="c31">Ma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font> = 
<font color="Maroon" title="c31">Ma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_2"><i><font color="Green" title="E92">A78</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="armstrng.html#E2:98_2_2_4_2"><i><font color="Green" title="E68">A52</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_2"><i><font color="Green" title="E72">A56</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3"><i><font color="Green" title="E84">A68</font></i></a>, <a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="txt" href="armstrng.html#E1:98_2_2_4_2_3_2_5_2"><i><font color="Green" title="E92">A78</font></i></a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Ma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">nj</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2"><i><font color="Green" title="E88">A72</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2_4_2_3_2_5_2"><i><font color="Green" title="E93">A79</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:98_2_2_4_2_3_2_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">j</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:98_2_2_4_2_3_2"><i><font color="Green" title="E91">A75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:98_2_2_4_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4_2_3"><i><font color="Green" title="E85">A69</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_2_3"><i><font color="Green" title="E86">A70</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><a NAME="E30:98_2_2_4_2"/><b>then </b>
<font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/><a NAME="E31:98_2_2_4_2"/><b>then </b><i><font color="Green" title="E82">A80</font></i>: 
<font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:98_2_2_4"><i><font color="Green" title="E66">A50</font></i></a>, <a class="txt" href="armstrng.html#E24:98_2_2_4_2"><i><font color="Green" title="E79">A63</font></i></a>, <a class="txt" href="armstrng.html#E28:98_2_2_4_2"><i><font color="Green" title="E81">A65</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>;<br/><b>set </b><font color="Maroon" title="c27">bs0</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>set </b><font color="Maroon" title="c28">bsi</font> = <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span>;<br/><i><font color="Green" title="E83">A81</font></i>:  <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">bx</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c25">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">bx</font>
<b>by </b><i><a class="txt" href="armstrng.html#E18:98_2_2_4_2"><i><font color="Green" title="E76">A60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">bx</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green" title="E70">A54</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">bx</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green" title="E70">A54</font></i></a>, <a class="txt" href="armstrng.html#E31:98_2_2_4_2"><i><font color="Green" title="E82">A80</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">bx</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E20:98_2_2_4_2"><i><font color="Green" title="E77">A61</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c29">Mbx</font> = <font color="Maroon" title="c8">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">bx</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c7">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/><a NAME="E34:98_2_2_4_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">Mbx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E84">A82</font></i>: <font color="Maroon" title="c29">Mbx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c29">Mbx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="armstrng.html#E6:98_2_2_4_2"><i><font color="Green" title="E70">A54</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="txt" href="armstrng.html#E10:98_2_2_4_2"><i><font color="Green" title="E72">A56</font></i></a></i>
;<br/><i><font color="Green" title="E85">A83</font></i>: <font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="binari_3.html#T26" title="BINARI_3:th.26">BINARI_3:26</a></i>
<br/>.= 
<font color="Maroon" title="c7">n</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="euclid.html#D4" title="EUCLID:def.4">EUCLID:def 4</a></i>
;<br/><a NAME="E37:98_2_2_4_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E86">A84</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="txt" href="armstrng.html#E36:98_2_2_4_2"><i><font color="Green" title="E85">A83</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green" title="E87">A85</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <font color="Maroon" title="c29">Mbx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">Mbx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E35:98_2_2_4_2"><i><font color="Green" title="E84">A82</font></i></a></i>
;<br/><i><font color="Green" title="E88">A86</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="armstrng.html#K3" title="ARMSTRNG:func.3">'&amp;'</a> <font color="Maroon" title="c29">Mbx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">Mbx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="armstrng.html#D5" target="_self" title="ARMSTRNG:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E35:98_2_2_4_2"><i><font color="Green" title="E84">A82</font></i></a></i>
;<br/><a NAME="E41:98_2_2_4_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="armstrng.html#E3:98_2_2"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="armstrng.html#E9:98_2_2_4_2"><i><font color="Green" title="E71">A55</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="armstrng.html#E14:98_2_2_4_2"><i><font color="Green" title="E74">A58</font></i></a>, <a class="txt" href="armstrng.html#E16:98_2_2_4_2"><i><font color="Green" title="E75">A59</font></i></a>, <a class="ref" href="armstrng.html#T8" target="_self" title="ARMSTRNG:th.8">Th8</a></i>
;<br/><b>hence </b><a NAME="E43:98_2_2_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E32:98_2_2_4_2"><i><font color="Green" title="E83">A81</font></i></a>, <a class="txt" href="armstrng.html#E38:98_2_2_4_2"><i><font color="Green" title="E86">A84</font></i></a>, <a class="txt" href="armstrng.html#E39:98_2_2_4_2"><i><font color="Green" title="E87">A85</font></i></a>, <a class="txt" href="armstrng.html#E40:98_2_2_4_2"><i><font color="Green" title="E88">A86</font></i></a>, <a class="ref" href="binari_3.html#T2" title="BINARI_3:th.2">BINARI_3:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:98_2_2_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:98"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="armstrng.html#E4:98_2_2_4"><i><font color="Green" title="E65">A49</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:98_2_2"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c6">D</font>,<font color="Maroon" title="c10">R</font> #)
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
