<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> &amp; <font color="Olive" title="b1">C</font> is <a href="armstrng.html#V11" title="ARMSTRNG:attr.11">without_proper_subsets</a> holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a>  st <br/>( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Olive" title="b2">R</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a> &amp; <font color="Maroon" title="c2">C</font> is <a href="armstrng.html#V11" title="ARMSTRNG:attr.11">without_proper_subsets</a> implies  ex <font color="Olive" title="b1">R</font> being   <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a>  st <br/>( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Olive" title="b1">R</font></span>)</span> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:107"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">(C1)</a>
 <b>and </b><br/><a NAME="E2:107"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="armstrng.html#V11" title="ARMSTRNG:attr.11">without_proper_subsets</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being   <a href="armstrng.html#L1" title="ARMSTRNG:struct.1">DB-Rel</a>  st <br/>( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Olive" title="b1">R</font></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c3">M</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span> ;<br/>
<a NAME="E3:107"/>
 <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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:107"/><b>then </b><i><font color="Green" title="E59">A3</font></i>: 
 <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">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">M0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> 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/>
<b>set </b><font color="Maroon" title="c6">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </span>}</span> ;<br/>
<a NAME="E7:107"/><i><font color="Green" title="E60">A4</font></i>: 
<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:107"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/>
<a NAME="E8:107"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </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="c5">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </span>}</span>  or <font color="Maroon" title="c7">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="c5">D</font> )</span><br/>

<b>assume </b><a NAME="E1:107_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">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="c5">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">L</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="E3:107_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span>
 <b>and </b><br/><a NAME="E4:107_1"/>
<font color="Maroon" title="c8">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 ;<br/>
<b>set </b><font color="Maroon" title="c9">g</font> = <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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span>;<br/>
<a NAME="E5:107_1"/><i><font color="Green" title="E62">A6</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="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">L</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then </b><i><font color="Green" title="E63">A7</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">L</font>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">L</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E7:107_1"/><i><font color="Green" title="E64">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">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><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_2"><b>now </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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">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="c5">D</font> implies <span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:107_1_2"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c10">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="c5">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:107_1_2"/><i><font color="Green" title="E66">A10</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:107_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="armstrng.html#E1:107_1_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">L</font> or  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_1_1"><b>suppose </b></a><a NAME="E1:107_1_2_1_1"/><i><font color="Green" title="E67">A11</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>then </b><span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:107_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c8">L</font>
<b>by </b><i><a class="txt" href="armstrng.html#E1:107_1_2_1_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:107_1_2_1_1"/>
<span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107_1_2"><i><font color="Green" title="E66">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_2_1_2"><b>suppose </b></a><a NAME="E1:107_1_2_1_2"/>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>then </b><span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<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="c10">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:107_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="txt" href="armstrng.html#E7:107_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="armstrng.html#E1:107_1_2"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:107_1_2_1_2"/>
<span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c8">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107_1_2"><i><font color="Green" title="E66">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:107_1"/>
<font color="Maroon" title="c7">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="c5">D</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:107_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="armstrng.html#E6:107_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="txt" href="armstrng.html#E7:107_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">R</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">L</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Olive" title="b1">L</font></span>)</span></span>)</span> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font> </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="c5">D</font></span>)</span> <b>by </b><i><a class="txt" href="armstrng.html#E7:107"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">DB</font> =  <a href="armstrng.html#G1" title="ARMSTRNG:aggr.1">DB-Rel</a>(# <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c7">R</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c8">DB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E10:107"/>
the <a href="armstrng.html#U1" title="ARMSTRNG:sel.1">Attributes</a> of <font color="Maroon" title="c8">DB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c9">ds</font> =  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font>;<br/>
<b>set </b><font color="Maroon" title="c10">ck</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span> ;<br/>
<a NAME="E11:107"/><i><font color="Green" title="E61">A12</font></i>: 
 <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<div><i><font color="Green" title="E62">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:107_2"/><i><font color="Green" title="E63">A14</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span> </span><br/><b>then reconsider </b><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c11">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c13">AA</font> = <font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c14">XA</font> = <font color="Maroon" title="c1">X</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 <font color="Maroon" title="c8">DB</font> <b>by </b><i><a class="txt" href="armstrng.html#E11:107"><i><font color="Green" title="E61">A12</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">f</font>, <font color="Maroon" title="c16">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 <font color="Maroon" title="c8">DB</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font> implies <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> )</span><br/><b>assume </b><a NAME="E1:107_2_1"/><i><font color="Green" title="E64">A15</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span><br/><a NAME="E2:107_2_1"/>
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">Lf</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="E4:107_2_1"/><i><font color="Green" title="E65">A16</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">Lf</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c17">Lf</font></span>)</span>
 <b>and </b><br/><a NAME="E5:107_2_1"/><i><font color="Green" title="E66">A17</font></i>: 
<font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 ;<br/><a NAME="E6:107_2_1"/>
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c18">Lg</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="E8:107_2_1"/><i><font color="Green" title="E67">A18</font></i>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Lg</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c18">Lg</font></span>)</span>
 <b>and </b><br/><a NAME="E9:107_2_1"/><i><font color="Green" title="E68">A19</font></i>: 
<font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 ;<br/><a NAME="E10:107_2_1"/><i><font color="Green" title="E69">A20</font></i>: 
( ( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  or <font color="Maroon" title="c17">Lf</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; ( <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  or <font color="Maroon" title="c18">Lg</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <b>by </b><i><a class="txt" href="armstrng.html#E5:107_2_1"><i><font color="Green" title="E66">A17</font></i></a>, <a class="txt" href="armstrng.html#E9:107_2_1"><i><font color="Green" title="E68">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:107_2_1"/><i><font color="Green" title="E70">A21</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="c18">Lg</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c18">Lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E12:107_2_1"/><i><font color="Green" title="E71">A22</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="c17">Lf</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c17">Lf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lf</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_2_1_1"/>
( ( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  ) or ( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  ) or ( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E10:107_2_1"><i><font color="Green" title="E69">A20</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_1"><b>suppose </b></a><a NAME="E1:107_2_1_1_1"/>
( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">Lff</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="E3:107_2_1_1_1"/><i><font color="Green" title="E72">A23</font></i>: 
<font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Lff</font>
 <b>and </b><br/><a NAME="E4:107_2_1_1_1"/><i><font color="Green" title="E73">A24</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Maroon" title="c19">Lff</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:107_2_1_1_1"/><i><font color="Green" title="E74">A25</font></i>: 
<font color="Maroon" title="c17">Lf</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2"><i><font color="Green" title="E63">A14</font></i></a>, <a class="txt" href="armstrng.html#E3:107_2_1_1_1"><i><font color="Green" title="E72">A23</font></i></a>, <a class="txt" href="armstrng.html#E4:107_2_1_1_1"><i><font color="Green" title="E73">A24</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:107_2_1_1_1"/><i><font color="Green" title="E75">A26</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Lf</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:107_2_1_1_1"/><i><font color="Green" title="E76">A27</font></i>: 
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Lf</font> &amp; <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_1"><i><font color="Green" title="E75">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E77">A28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> = 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Lf</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c17">Lf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:107_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="txt" href="armstrng.html#E12:107_2_1"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_1_1_1"><i><font color="Green" title="E76">A27</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c17">Lf</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_1"><i><font color="Green" title="E76">A27</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><a NAME="E10:107_2_1_1_1"/><i><font color="Green" title="E78">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_1"><i><font color="Green" title="E76">A27</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E11:107_2_1_1_1"/>
( <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_2_1_1_1_2_1"/>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font> or  not <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:107_2_1_1_1_2_1_1"/><i><font color="Green" title="E79">A30</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font> )</span><br/><div class="add"><a NAME="E2:107_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Lg</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c18">Lg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="armstrng.html#E11:107_2_1"><i><font color="Green" title="E70">A21</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:107_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1_1_1_2_1_1"><i><font color="Green" title="E79">A30</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:107_2_1_1_1_2_1_2"/>
not <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font> )</span><br/><div class="add"><a NAME="E2:107_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="armstrng.html#E11:107_2_1"><i><font color="Green" title="E70">A21</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:107_2_1_1_1_2_1_2"/>
( <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Lg</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_1"><i><font color="Green" title="E75">A26</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E12:107_2_1_1_1"/>
<font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="armstrng.html#E4:107_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="armstrng.html#E5:107_2_1_1_1"><i><font color="Green" title="E74">A25</font></i></a>, <a class="txt" href="armstrng.html#E9:107_2_1_1_1"><i><font color="Green" title="E77">A28</font></i></a>, <a class="txt" href="armstrng.html#E10:107_2_1_1_1"><i><font color="Green" title="E78">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_2"><b>suppose </b></a><a NAME="E1:107_2_1_1_2"/><i><font color="Green" title="E72">A31</font></i>: 
( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">L</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="E3:107_2_1_1_2"/><i><font color="Green" title="E73">A32</font></i>: 
<font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">L</font>
 <b>and </b><br/><a NAME="E4:107_2_1_1_2"/><i><font color="Green" title="E74">A33</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Maroon" title="c19">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:107_2_1_1_2"/><i><font color="Green" title="E75">A34</font></i>: 
<font color="Maroon" title="c17">Lf</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2"><i><font color="Green" title="E63">A14</font></i></a>, <a class="txt" href="armstrng.html#E3:107_2_1_1_2"><i><font color="Green" title="E73">A32</font></i></a>, <a class="txt" href="armstrng.html#E4:107_2_1_1_2"><i><font color="Green" title="E74">A33</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:107_2_1_1_2"/><i><font color="Green" title="E76">A35</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Lf</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:107_2_1_1_2"/><i><font color="Green" title="E77">A36</font></i>: 
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Lf</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_2"><i><font color="Green" title="E76">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:107_2_1_1_2"/><i><font color="Green" title="E78">A37</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="c18">Lg</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c18">Lg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1_1_2"><i><font color="Green" title="E72">A31</font></i></a>, <a class="ref" href="funcop_1.html#T16" title="FUNCOP_1:th.16">FUNCOP_1:16</a></i>;<br/><i><font color="Green" title="E79">A38</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> = 
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_2"><i><font color="Green" title="E77">A36</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="armstrng.html#E9:107_2_1_1_2"><i><font color="Green" title="E78">A37</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>
<br/>.= 
<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="c20">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_2"><i><font color="Green" title="E76">A35</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> = 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_2"><i><font color="Green" title="E77">A36</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">Lf</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c17">Lf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:107_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="txt" href="armstrng.html#E12:107_2_1"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_1_1_2"><i><font color="Green" title="E77">A36</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c17">Lf</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_2"><i><font color="Green" title="E77">A36</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E12:107_2_1_1_2"/>
<font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="armstrng.html#E5:107_2_1_1_2"><i><font color="Green" title="E75">A34</font></i></a>, <a class="txt" href="armstrng.html#E10:107_2_1_1_2"><i><font color="Green" title="E79">A38</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_3"><b>suppose </b></a><a NAME="E1:107_2_1_1_3"/><i><font color="Green" title="E72">A39</font></i>: 
( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">L</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="E3:107_2_1_1_3"/><i><font color="Green" title="E73">A40</font></i>: 
<font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">L</font>
 <b>and </b><br/><a NAME="E4:107_2_1_1_3"/><i><font color="Green" title="E74">A41</font></i>: 
for <font color="Olive" title="b1">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Maroon" title="c19">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:107_2_1_1_3"/><i><font color="Green" title="E75">A42</font></i>: 
<font color="Maroon" title="c18">Lg</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2"><i><font color="Green" title="E63">A14</font></i></a>, <a class="txt" href="armstrng.html#E3:107_2_1_1_3"><i><font color="Green" title="E73">A40</font></i></a>, <a class="txt" href="armstrng.html#E4:107_2_1_1_3"><i><font color="Green" title="E74">A41</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:107_2_1_1_3"/><i><font color="Green" title="E76">A43</font></i>: 
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:107_2_1_1_3"/><i><font color="Green" title="E77">A44</font></i>: 
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Lg</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_3"><i><font color="Green" title="E76">A43</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:107_2_1_1_3"/><i><font color="Green" title="E78">A45</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="c17">Lf</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c17">Lf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1_1_3"><i><font color="Green" title="E72">A39</font></i></a>, <a class="ref" href="funcop_1.html#T16" title="FUNCOP_1:th.16">FUNCOP_1:16</a></i>;<br/><i><font color="Green" title="E79">A46</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> = 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_3"><i><font color="Green" title="E77">A44</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<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> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:107_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="txt" href="armstrng.html#E9:107_2_1_1_3"><i><font color="Green" title="E78">A45</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>
<br/>.= 
<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="c20">a</font>
<b>by </b><i><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E7:107_2_1_1_3"><i><font color="Green" title="E76">A43</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> = 
<font color="Maroon" title="c16">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_3"><i><font color="Green" title="E77">A44</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Lg</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c18">Lg</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a>, <a class="txt" href="armstrng.html#E11:107_2_1"><i><font color="Green" title="E70">A21</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_1_1_3"><i><font color="Green" title="E77">A44</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c18">Lg</font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:107_2_1_1_3"><i><font color="Green" title="E77">A44</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E12:107_2_1_1_3"/>
<font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_1"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="armstrng.html#E5:107_2_1_1_3"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="armstrng.html#E10:107_2_1_1_3"><i><font color="Green" title="E79">A46</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_2_1_1_4"><b>suppose </b></a><a NAME="E1:107_2_1_1_4"/>
( <font color="Maroon" title="c17">Lf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">Lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:107_2_1_1_4"/>
<font color="Maroon" title="c15">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:107_2_1"><i><font color="Green" title="E65">A16</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_1"><i><font color="Green" title="E67">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:107_2"/><b>then </b>
<font color="Maroon" title="c13">AA</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c14">XA</font>,<font color="Maroon" title="c8">DB</font>
 <b>by </b><i><a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>;<br/><a NAME="E6:107_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">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="E8:107_2"/><i><font color="Green" title="E64">A47</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span>
 <b>and </b><br/><a NAME="E9:107_2"/><i><font color="Green" title="E65">A48</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#R3" title="ARMSTRNG:pred.3">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T28" target="_self" title="ARMSTRNG:th.28">Th28</a></i>;<br/><a NAME="E10:107_2"/><i><font color="Green" title="E66">A49</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">b</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E9:107_2"><i><font color="Green" title="E65">A48</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><a NAME="E11:107_2"/><b>then </b><i><font color="Green" title="E67">A50</font></i>: 
<font color="Maroon" title="c16">b</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="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E12:107_2"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:107_2"><i><font color="Green" title="E64">A47</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">aa</font>, <font color="Maroon" title="c18">XX</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 <font color="Maroon" title="c8">DB</font><b> such that </b><br/><a NAME="E14:107_2"/><i><font color="Green" title="E68">A51</font></i>: 
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <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="c17">aa</font>,<font color="Maroon" title="c18">XX</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E15:107_2"/><i><font color="Green" title="E69">A52</font></i>: 
<font color="Maroon" title="c17">aa</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c18">XX</font>,<font color="Maroon" title="c8">DB</font>
 ;<br/><a NAME="E16:107_2"/><i><font color="Green" title="E70">A53</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">aa</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">XX</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E14:107_2"><i><font color="Green" title="E68">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_2_2"/><i><font color="Green" title="E71">A54</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>set </b><font color="Maroon" title="c19">r0</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>set </b><font color="Maroon" title="c20">r1</font> = <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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>;<br/><a NAME="E2:107_2_2"/>
 <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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:107_2_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:107_2_2"/><b>then </b><i><font color="Green" title="E72">A55</font></i>: 
<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies  not <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>assume </b><a NAME="E1:107_2_2_1"/><i><font color="Green" title="E73">A56</font></i>: 
<font color="Maroon" title="c21">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><b>assume </b><a NAME="E2:107_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E3:107_2_2_1"/><b>then </b>
<font color="Maroon" title="c21">K</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:107_2_2_1"/><b>then </b><i><font color="Green" title="E74">A57</font></i>: 
<font color="Maroon" title="c21">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/><a NAME="E5:107_2_2_1"/><b>then </b>
<font color="Maroon" title="c21">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:107_2"><i><font color="Green" title="E66">A49</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:107_2_2_1"/><b>then </b>
<font color="Maroon" title="c21">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="armstrng.html#E1:107_2"><i><font color="Green" title="E63">A14</font></i></a>, <a class="txt" href="armstrng.html#E1:107_2_2_1"><i><font color="Green" title="E73">A56</font></i></a>, <a class="ref" href="armstrng.html#D28" target="_self" title="ARMSTRNG:def.28">Def28</a></i>;<br/><b>hence </b><a NAME="E7:107_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E10:107_2"><i><font color="Green" title="E66">A49</font></i></a>, <a class="txt" href="armstrng.html#E1:107_2_2"><i><font color="Green" title="E71">A54</font></i></a>, <a class="txt" href="armstrng.html#E4:107_2_2_1"><i><font color="Green" title="E74">A57</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:107_2_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span> 
 ;<br/><a NAME="E7:107_2_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:107_2_2"/><b>then </b><i><font color="Green" title="E73">A58</font></i>: 
<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><a NAME="E9:107_2_2"/><i><font color="Green" title="E74">A59</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">r0</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> , <font color="Maroon" title="c22">r1</font> = <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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> as    <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 <font color="Maroon" title="c8">DB</font> <b>by </b><i><a class="txt" href="armstrng.html#E4:107_2_2"><i><font color="Green" title="E72">A55</font></i></a>, <a class="txt" href="armstrng.html#E8:107_2_2"><i><font color="Green" title="E73">A58</font></i></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2_2"><b>now </b></a><div class="add"><a NAME="E1:107_2_2_2"/><i><font color="Green" title="E75">A60</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="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">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="c21">r0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E2:107_2_2_2"/><i><font color="Green" title="E76">A61</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">r0</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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">r1</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/><b>hence </b><a NAME="E4:107_2_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">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="c22">r1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107_2_2_2"><i><font color="Green" title="E76">A61</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> <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> <font color="Maroon" title="c22">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">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="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )</span><br/><b>assume </b><a NAME="E5:107_2_2_2"/>
<font color="Maroon" title="c23">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="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span><br/><a NAME="E6:107_2_2_2"/><b>then </b><i><font color="Green" title="E77">A62</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_2_2_2"><i><font color="Green" title="E75">A60</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:107_2_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E8:107_2_2_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E9:107_2_2_2"/><b>then </b><i><font color="Green" title="E78">A63</font></i>: 
not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E6:107_2_2_2"><i><font color="Green" title="E77">A62</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> = 
<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="c23">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:107_2_2_2"><i><font color="Green" title="E77">A62</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="c22">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E9:107_2_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="txt" href="armstrng.html#E9:107_2_2_2"><i><font color="Green" title="E78">A63</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E12:107_2_2"/><b>then </b><i><font color="Green" title="E75">A64</font></i>: 
<font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c15">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/><div><i><font color="Green" title="E76">A65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_2_2_3"/>
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:107_2_2_3"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:107_2_2_3"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:107"><i><font color="Green" title="E61">A12</font></i></a>, <a class="ref" href="subset_1.html#T39" title="SUBSET_1:th.39">SUBSET_1:39</a></i>;<br/><b>hence </b><a NAME="E4:107_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E10:107_2"><i><font color="Green" title="E66">A49</font></i></a>, <a class="txt" href="armstrng.html#E1:107_2_2"><i><font color="Green" title="E71">A54</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c23">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:107_2_2"/><i><font color="Green" title="E77">A66</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><i><font color="Green" title="E78">A67</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c21">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> = 
<font color="Maroon" title="c21">r0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="armstrng.html#E15:107_2_2"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E15:107_2_2"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c22">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> = 
<font color="Maroon" title="c22">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="armstrng.html#E15:107_2_2"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
<b>by </b><i><a class="txt" href="armstrng.html#E9:107_2_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="txt" href="armstrng.html#E15:107_2_2"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c15">a</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E15:107_2_2"><i><font color="Green" title="E77">A66</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E18:107_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E11:107_2"><i><font color="Green" title="E67">A50</font></i></a>, <a class="txt" href="armstrng.html#E15:107_2"><i><font color="Green" title="E69">A52</font></i></a>, <a class="txt" href="armstrng.html#E16:107_2"><i><font color="Green" title="E70">A53</font></i></a>, <a class="txt" href="armstrng.html#E12:107_2_2"><i><font color="Green" title="E75">A64</font></i></a>, <a class="txt" href="armstrng.html#E13:107_2_2"><i><font color="Green" title="E76">A65</font></i></a>, <a class="txt" href="armstrng.html#E16:107_2_2"><i><font color="Green" title="E78">A67</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E18:107_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:107_2"><i><font color="Green" title="E64">A47</font></i></a>, <a class="txt" href="armstrng.html#E10:107_2"><i><font color="Green" title="E66">A49</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E19:107_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span>  ) &amp; ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> ) )</span><br/><b>thus </b><a NAME="E1:107_3"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span>  )
 <b>by </b><i><a class="txt" href="armstrng.html#E12:107"><i><font color="Green" title="E62">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> )</span><br/><b>assume </b><a NAME="E2:107_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font></span><br/><b>then consider </b><font color="Maroon" title="c12">A</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="E4:107_3"/><i><font color="Green" title="E63">A68</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font>
 <b>and </b><br/><a NAME="E5:107_3"/><i><font color="Green" title="E64">A69</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span>
 ;<br/><a NAME="E6:107_3"/><i><font color="Green" title="E65">A70</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:107_3"><i><font color="Green" title="E64">A69</font></i></a></i>;<br/><div><i><font color="Green" title="E66">A71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">K</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Maroon" title="c13">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font> implies not <font color="Maroon" title="c13">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">A</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:107_3_1"/><i><font color="Green" title="E67">A72</font></i>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>and </b><br/><a NAME="E2:107_3_1"/><i><font color="Green" title="E68">A73</font></i>: 
<font color="Maroon" title="c13">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c13">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">A</font></span><br/><a NAME="E3:107_3_1"/>
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E12:107"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="armstrng.html#E1:107_3_1"><i><font color="Green" title="E67">A72</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">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="E5:107_3_1"/><i><font color="Green" title="E69">A74</font></i>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font>
 <b>and </b><br/><a NAME="E6:107_3_1"/><i><font color="Green" title="E70">A75</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">B</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="armstrng.html#K8" title="ARMSTRNG:func.8">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">AA</font> = <font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c16">B</font> = <font color="Maroon" title="c14">B</font>, <font color="Maroon" title="c17">XA</font> = <font color="Maroon" title="c1">X</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 <font color="Maroon" title="c8">DB</font> <b>by </b><i><a class="txt" href="armstrng.html#E11:107"><i><font color="Green" title="E61">A12</font></i></a></i>;<br/><b>assume </b><a NAME="E8:107_3_1"/><i><font color="Green" title="E71">A76</font></i>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E9:107_3_1"/><i><font color="Green" title="E72">A77</font></i>: 
<font color="Maroon" title="c12">A</font> <a href="armstrng.html#R4" title="ARMSTRNG:pred.4">^|^</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c1">X</font>, <a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:107_3"><i><font color="Green" title="E64">A69</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self" title="ARMSTRNG:def.18">Def18</a></i>;<br/><a NAME="E10:107_3_1"/>
<span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c15">AA</font>,<font color="Maroon" title="c17">XA</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span> <a href="armstrng.html#NR6" title="ARMSTRNG:NR.6">&lt;=</a> <span class="p1"><a href="armstrng.html#K6" title="ARMSTRNG:func.6">[</a><span class="default"><font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c17">XA</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107_3_1"><i><font color="Green" title="E68">A73</font></i></a>, <a class="txt" href="armstrng.html#E5:107_3_1"><i><font color="Green" title="E69">A74</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self" title="ARMSTRNG:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E11:107_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E5:107_3_1"><i><font color="Green" title="E69">A74</font></i></a>, <a class="txt" href="armstrng.html#E6:107_3_1"><i><font color="Green" title="E70">A75</font></i></a>, <a class="txt" href="armstrng.html#E8:107_3_1"><i><font color="Green" title="E71">A76</font></i></a>, <a class="txt" href="armstrng.html#E9:107_3_1"><i><font color="Green" title="E72">A77</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self" title="ARMSTRNG:th.29">Th29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c13">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:107_3"/><i><font color="Green" title="E67">A78</font></i>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">K</font> = <font color="Maroon" title="c13">K</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="armstrng.html#E9:107_3"><i><font color="Green" title="E67">A78</font></i></a></i>;<br/><b>assume </b><a NAME="E11:107_3"/><i><font color="Green" title="E68">A79</font></i>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E12:107_3"/><b>then </b>
not <font color="Maroon" title="c14">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:107_3"><i><font color="Green" title="E63">A68</font></i></a>, <a class="txt" href="armstrng.html#E7:107_3"><i><font color="Green" title="E66">A71</font></i></a>, <a class="txt" href="armstrng.html#E9:107_3"><i><font color="Green" title="E67">A78</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:107_3"/><i><font color="Green" title="E69">A80</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">K</font> &amp; not <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c16">m</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span> ;<br/><b>reconsider </b><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c15">k</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#E14:107_3"><i><font color="Green" title="E69">A80</font></i></a></i>;<br/><a NAME="E16:107_3"/><i><font color="Green" title="E70">A81</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E9:107_3"><i><font color="Green" title="E67">A78</font></i></a>, <a class="txt" href="armstrng.html#E14:107_3"><i><font color="Green" title="E69">A80</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:107_3"/><i><font color="Green" title="E71">A82</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span>  implies <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/><b>assume </b><a NAME="E1:107_3_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><a NAME="E2:107_3_2"/><b>then </b>
 ex <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> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) )
 ;<br/><b>hence </b><a NAME="E3:107_3_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c19">m</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">K</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> implies <font color="Maroon" title="c19">m</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><b>assume </b><a NAME="E1:107_3_3"/><i><font color="Green" title="E72">A83</font></i>: 
<font color="Maroon" title="c20">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">m</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E2:107_3_3"/>
not <font color="Maroon" title="c20">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:107_3"><i><font color="Green" title="E63">A68</font></i></a>, <a class="txt" href="armstrng.html#E7:107_3"><i><font color="Green" title="E66">A71</font></i></a>, <a class="txt" href="armstrng.html#E11:107_3"><i><font color="Green" title="E68">A79</font></i></a>, <a class="txt" href="armstrng.html#E1:107_3_3"><i><font color="Green" title="E72">A83</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:107_3_3"/><i><font color="Green" title="E73">A84</font></i>: 
( <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">K</font> &amp; not <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:107_3_3"/>
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">m</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_3_3"><i><font color="Green" title="E72">A83</font></i></a>, <a class="txt" href="armstrng.html#E4:107_3_3"><i><font color="Green" title="E73">A84</font></i></a></i>;<br/><b>hence </b><a NAME="E6:107_3_3"/>
<font color="Maroon" title="c19">m</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:107_3_3"><i><font color="Green" title="E73">A84</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E22:107_3"/><b>then </b><i><font color="Green" title="E72">A85</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">L</font> where <font color="Olive" title="b1">L</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : for <font color="Olive" title="b2">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> holds <br/><font color="Olive" title="b1">L</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  </span>}</span> 
 ;<br/><b>set </b><font color="Maroon" title="c20">r0</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>set </b><font color="Maroon" title="c21">r1</font> = <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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span>;<br/><a NAME="E23:107_3"/>
 <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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E24:107_3"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E25:107_3"/><b>then </b><i><font color="Green" title="E73">A86</font></i>: 
<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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><a NAME="E26:107_3"/>
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M0</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E22:107_3"><i><font color="Green" title="E72">A85</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E27:107_3"/><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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c22">r0</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> , <font color="Maroon" title="c23">r1</font> = <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_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span> as    <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 <font color="Maroon" title="c8">DB</font> <b>by </b><i><a class="txt" href="armstrng.html#E25:107_3"><i><font color="Green" title="E73">A86</font></i></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/><a NAME="E29:107_3"/><i><font color="Green" title="E74">A87</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">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">m</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_4"><b>now </b></a><div class="add"><a NAME="E1:107_3_4"/><i><font color="Green" title="E75">A88</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="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">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="c22">r0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E2:107_3_4"/><i><font color="Green" title="E76">A89</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c22">r0</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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">r1</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <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>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c19">m</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/><b>hence </b><a NAME="E4:107_3_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">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="c23">r1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:107_3_4"><i><font color="Green" title="E76">A89</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <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> <font color="Maroon" title="c23">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">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="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )</span><br/><b>assume </b><a NAME="E5:107_3_4"/><i><font color="Green" title="E77">A90</font></i>: 
<font color="Maroon" title="c24">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="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span><br/><a NAME="E6:107_3_4"/><i><font color="Green" title="E78">A91</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_3_4"><i><font color="Green" title="E75">A88</font></i></a>, <a class="txt" href="armstrng.html#E5:107_3_4"><i><font color="Green" title="E77">A90</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green" title="E79">A92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:107_3_4_2"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:107_3_4_2"/><b>then </b>
 ex <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> st <br/>( <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font> &amp;  ex <font color="Olive" title="b2">K</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> ) )
 ;<br/><b>hence </b><a NAME="E3:107_3_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E1:107_3_4"><i><font color="Green" title="E75">A88</font></i></a>, <a class="txt" href="armstrng.html#E5:107_3_4"><i><font color="Green" title="E77">A90</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<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="c24">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:107_3_4"><i><font color="Green" title="E78">A91</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="c23">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<b>by </b><i><a class="txt" href="armstrng.html#E29:107_3"><i><font color="Green" title="E74">A87</font></i></a>, <a class="txt" href="armstrng.html#E7:107_3_4"><i><font color="Green" title="E79">A92</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E31:107_3"/><b>then </b><i><font color="Green" title="E75">A93</font></i>: 
<font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">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/><b>consider </b><font color="Maroon" title="c24">aa</font>, <font color="Maroon" title="c25">XX</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 <font color="Maroon" title="c8">DB</font><b> such that </b><br/><a NAME="E33:107_3"/><i><font color="Green" title="E76">A94</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c1">X</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c24">aa</font>,<font color="Maroon" title="c25">XX</font></span><a href="armstrng.html#K6" title="ARMSTRNG:func.6">]</a></span>
 <b>and </b><br/><a NAME="E34:107_3"/><i><font color="Green" title="E77">A95</font></i>: 
<font color="Maroon" title="c24">aa</font> <a href="armstrng.html#R2" title="ARMSTRNG:pred.2">&gt;|&gt;</a> <font color="Maroon" title="c25">XX</font>,<font color="Maroon" title="c8">DB</font>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:107_3"><i><font color="Green" title="E65">A70</font></i></a></i>;<br/><a NAME="E35:107_3"/><i><font color="Green" title="E78">A96</font></i>: 
( <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">aa</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">XX</font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E33:107_3"><i><font color="Green" title="E76">A94</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E36:107_3"/><i><font color="Green" title="E79">A97</font></i>: 
<font color="Maroon" title="c19">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 ;<br/><b>then </b><i><font color="Green" title="E80">A98</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c22">r0</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> = 
<font color="Maroon" title="c22">r0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="armstrng.html#E18:107_3"><i><font color="Green" title="E71">A82</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E18:107_3"><i><font color="Green" title="E71">A82</font></i></a>, <a class="txt" href="armstrng.html#E36:107_3"><i><font color="Green" title="E79">A97</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> = 
<font color="Maroon" title="c23">r1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="armstrng.html#E18:107_3"><i><font color="Green" title="E71">A82</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">m</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
<b>by </b><i><a class="txt" href="armstrng.html#E18:107_3"><i><font color="Green" title="E71">A82</font></i></a>, <a class="txt" href="armstrng.html#E29:107_3"><i><font color="Green" title="E74">A87</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c19">m</font>
<b>by </b><i><a class="txt" href="armstrng.html#E18:107_3"><i><font color="Green" title="E71">A82</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E39:107_3"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E16:107_3"><i><font color="Green" title="E70">A81</font></i></a>, <a class="txt" href="armstrng.html#E31:107_3"><i><font color="Green" title="E75">A93</font></i></a>, <a class="txt" href="armstrng.html#E34:107_3"><i><font color="Green" title="E77">A95</font></i></a>, <a class="txt" href="armstrng.html#E35:107_3"><i><font color="Green" title="E78">A96</font></i></a>, <a class="txt" href="armstrng.html#E37:107_3"><i><font color="Green" title="E80">A98</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self" title="ARMSTRNG:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:107"/>
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="armstrng.html#K13" title="ARMSTRNG:func.13">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" title="ARMSTRNG:func.5">Dependency-str</a> <font color="Maroon" title="c8">DB</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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