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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">(C1)</a>
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="armstrng.html#V11" target="_self">without_proper_subsets</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span> ;<br/>
<a NAME="E3:108"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:108"/><b>then </b><i><font color="Green">E60</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  <a href="relat_1.html#V2">V12</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/>
<a NAME="E7:108"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:108"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E8:108"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:108_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:108_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:108_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/>
<a NAME="E5:108_1"/><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then </b><i><font color="Green">E64</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E7:108_1"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:108_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:108_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E1:108_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1_1"><b>suppose </b></a><a NAME="E1:108_1_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:108_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E1:108_1_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:108_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108_1_2"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1_2"><b>suppose </b></a><a NAME="E1:108_1_2_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E5:108_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E7:108_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E1:108_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:108_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108_1_2"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:108_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E3:108_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="armstrng.html#E6:108_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E7:108_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="txt" href="armstrng.html#E7:108"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>8</sub></font> =  <a href="armstrng.html#G1" target="_self">DB-Rel</a>(# <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> #);<br/>

<b>thus </b><a NAME="E10:108"/>
the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span> ;<br/>
<a NAME="E11:108"/><i><font color="Green">E62</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:108_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E11:108"><i><font color="Green">E62</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>assume </b><a NAME="E1:108_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E2:108_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:108_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:108_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E6:108_2_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:108_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:108_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E10:108_2_1"/><i><font color="Green">E70</font></i>: 
( ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> ) )
 <b>by </b><i><a class="txt" href="armstrng.html#E5:108_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="armstrng.html#E9:108_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:108_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E12:108_2_1"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_2_1_1"/>
( ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  ) or ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  ) or ( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="armstrng.html#E10:108_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_1"><b>suppose </b></a><a NAME="E1:108_2_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:108_2_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E4:108_2_1_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E5:108_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E3:108_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E4:108_2_1_1_1"><i><font color="Green">E74</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:108_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:108_2_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then </b><i><font color="Green">E78</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:108_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E12:108_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E10:108_2_1_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E11:108_2_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_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="E1:108_2_1_1_1_2_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> or  not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:108_2_1_1_1_2_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:108_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E11:108_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:108_2_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1_1_1_2_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:108_2_1_1_1_2_1_2"/>
not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><div class="add"><a NAME="E2:108_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E11:108_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:108_2_1_1_1_2_1_2"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E12:108_2_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E4:108_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E5:108_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="armstrng.html#E9:108_2_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="armstrng.html#E10:108_2_1_1_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_2"><b>suppose </b></a><a NAME="E1:108_2_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:108_2_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E4:108_2_1_1_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E5:108_2_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E3:108_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="armstrng.html#E4:108_2_1_1_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:108_2_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:108_2_1_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:108_2_1_1_2"/><i><font color="Green">E79</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/><i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E9:108_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:108_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E12:108_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E12:108_2_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E5:108_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E10:108_2_1_1_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_3"><b>suppose </b></a><a NAME="E1:108_2_1_1_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:108_2_1_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E4:108_2_1_1_3"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E5:108_2_1_1_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E3:108_2_1_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="armstrng.html#E4:108_2_1_1_3"><i><font color="Green">E75</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:108_2_1_1_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E8:108_2_1_1_3"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:108_2_1_1_3"/><i><font color="Green">E79</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/><i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E4:108_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E9:108_2_1_1_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E7:108_2_1_1_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E11:108_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_1_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E8:108_2_1_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E12:108_2_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E5:108_2_1_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E10:108_2_1_1_3"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1_1_4"><b>suppose </b></a><a NAME="E1:108_2_1_1_4"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_2_1_1_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:108_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:108_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="armstrng.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E6:108_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:108_2"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:108_2"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="armstrng.html#R3" target="_self">&gt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="armstrng.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E10:108_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:108_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E11:108_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E9:108_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E12:108_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E13:108_2"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E10:108_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E15:108_2"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E16:108_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E17:108_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E15:108_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/><a NAME="E2:108_2_2"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:108_2_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:108_2_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:108_2_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>assume </b><a NAME="E2:108_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:108_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:108_2_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><a NAME="E5:108_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:108_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:108_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108"><i><font color="Green">E59</font></i></a>, <a class="txt" href="armstrng.html#E1:108_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E1:108_2_2_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="armstrng.html#D28" target="_self">Def28</a></i>;<br/><b>hence </b><a NAME="E7:108_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E11:108_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E1:108_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E4:108_2_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E6:108_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span> 
 ;<br/><a NAME="E7:108_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:108_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E9:108_2_2"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0, <font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E4:108_2_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="armstrng.html#E8:108_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2_2"><b>now </b></a><div class="add"><a NAME="E1:108_2_2_2"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E2:108_2_2_2"/><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><b>hence </b><a NAME="E4:108_2_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108_2_2_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E5:108_2_2_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 ;<br/><a NAME="E6:108_2_2_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_2_2_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:108_2_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E8:108_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E9:108_2_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E6:108_2_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:108_2_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E9:108_2_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E9:108_2_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div><a NAME="E12:108_2_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2_2_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:108_2_2_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:108_2_2_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E11:108"><i><font color="Green">E62</font></i></a>, <a class="ref" href="subset_1.html#T39">SUBSET_1:39</a></i>;<br/><b>hence </b><a NAME="E4:108_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E11:108_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E1:108_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:108_2_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E15:108_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E15:108_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E15:108_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E9:108_2_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E15:108_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="txt" href="armstrng.html#E15:108_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E18:108_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E12:108_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="armstrng.html#E16:108_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E17:108_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E12:108_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="armstrng.html#E13:108_2_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="armstrng.html#E16:108_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><a NAME="E19:108_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E8:108_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="armstrng.html#E11:108_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E20:108_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span> 
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:108_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span>  )
 <b>by </b><i><a class="txt" href="armstrng.html#E12:108"><i><font color="Green">E63</font></i></a></i>;<br/><b>assume </b><a NAME="E2:108_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:108_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E5:108_3"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E6:108_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:108_3"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E7:108_3"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:108_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:108_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E3:108_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E12:108"><i><font color="Green">E63</font></i></a>, <a class="txt" href="armstrng.html#E1:108_3_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:108_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E6:108_3_1"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E11:108"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E8:108_3_1"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K7" target="_self">Dependencies-Order</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="armstrng.html#K1" target="_self">Maximal_in</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="armstrng.html#E6:108_3_1"><i><font color="Green">E71</font></i></a></i>;<br/><b>assume </b><a NAME="E9:108_3_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E10:108_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="armstrng.html#R4" target="_self">^|^</a>  <a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>, <a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E5:108_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="armstrng.html#D18" target="_self">Def18</a></i>;<br/><a NAME="E11:108_3_1"/>
<span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span> <a href="armstrng.html#NR6" target="_self">&lt;=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="armstrng.html#E5:108_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="armstrng.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E12:108_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E5:108_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="armstrng.html#E8:108_3_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E9:108_3_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E10:108_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="armstrng.html#T29" target="_self">Th29</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:108_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E10:108_3"><i><font color="Green">E68</font></i></a></i>;<br/><b>assume </b><a NAME="E12:108_3"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E13:108_3"/><b>then </b>
not <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:108_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E8:108_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="armstrng.html#E10:108_3"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:108_3"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E15:108_3"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E17:108_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="armstrng.html#E10:108_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="armstrng.html#E15:108_3"><i><font color="Green">E70</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:108_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:108_3_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 ;<br/><a NAME="E2:108_3_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>hence </b><a NAME="E3:108_3_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:108_3_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:108_3_3"/>
not <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E4:108_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E8:108_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="armstrng.html#E12:108_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="armstrng.html#E1:108_3_3"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:108_3_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> &amp; not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:108_3_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_3_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="armstrng.html#E4:108_3_3"><i><font color="Green">E74</font></i></a></i>;<br/><b>hence </b><a NAME="E6:108_3_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="armstrng.html#E4:108_3_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E23:108_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span> 
 ;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>;<br/><a NAME="E24:108_3"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E25:108_3"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E26:108_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E27:108_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E28:108_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E23:108_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E29:108_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0, <font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of the <a href="armstrng.html#U3" target="_self">Relationship</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E27:108_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/><a NAME="E31:108_3"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_4"><b>now </b></a><div class="add"><a NAME="E1:108_3_4"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E2:108_3_4"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><b>hence </b><a NAME="E4:108_3_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E2:108_3_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E5:108_3_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/><a NAME="E6:108_3_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_3_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E5:108_3_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_3_4_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:108_3_4_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>hence </b><a NAME="E3:108_3_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E1:108_3_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E5:108_3_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="margrel1.html#K1">--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E6:108_3_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E31:108_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="armstrng.html#E7:108_3_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/></div><b>end;</b></div><a NAME="E33:108_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="armstrng.html#U1" target="_self">Attributes</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E35:108_3"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="armstrng.html#K6" target="_self">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="armstrng.html#K6" target="_self">]</a></span>
 <b>and </b><br/><a NAME="E36:108_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="armstrng.html#R2" target="_self">&gt;|&gt;</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="armstrng.html#E7:108_3"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E37:108_3"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="armstrng.html#E35:108_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E38:108_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then </b><i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E19:108_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="armstrng.html#E19:108_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E38:108_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E19:108_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E19:108_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="armstrng.html#E31:108_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="armstrng.html#E19:108_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E41:108_3"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E17:108_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="armstrng.html#E33:108_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="armstrng.html#E36:108_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="armstrng.html#E37:108_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="armstrng.html#E39:108_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="armstrng.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:108"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K8" target="_self">Maximal_wrt</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E15:108"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="armstrng.html#K13" target="_self">candidate-keys</a> <span class="p1">(<span class="default"><a href="armstrng.html#K5" target="_self">Dependency-str</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>


</div>
