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

<b>let </b><font color="Maroon">C</font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>;<br/>



<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<b>set </b><font color="Maroon">OU</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1"/>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><b>case </b></a><a NAME="E1:5_1_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:5_1_1_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:5_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E2:5_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">o'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:5_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">x'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E6:5_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E2:5_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:5_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:5_1_1_1_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:5_1_1_1_1_1_2"/><b>then </b>
for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:5_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:5_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:5_1_1_1_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</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"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_1_3"/>
<font color="Maroon">Y</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"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:5_1_1_1_1_1_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/>
<b>thus </b><a NAME="E4:5_1_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:5_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:5_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> ;<br/>
<a NAME="E4:5_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">d</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"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:5_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:5_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> <b>by </b><i>, </i>;<br/>
<a NAME="E9:5_1_1_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:5_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</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"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A'</font></span>)</span> where <font color="Olive">A'</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:5_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:5_1_1_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E2:5_1_1_1_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:5_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:5_1_1_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o</font> where <font color="Olive">o</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:5_1_1_1_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">x'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E7:5_1_1_1_1_3"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:5_1_1_1_1"/>
for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_1_1_1_1_4"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:5_1_1_1_1_4"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:5_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">o'</font> where <font color="Olive">o'</font> is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> : for <font color="Olive">a</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">o'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E13:5_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2"><b>case </b></a><a NAME="E1:5_1_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:5_1_1_2"/>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1_1_2_1"/>
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font>;<br/>
<a NAME="E3:5_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:5_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:5_1_1_2"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <font color="Maroon">C</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 ;<br/>


</div>
