<?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">O</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>;<br/>



<a NAME="E1:28"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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#D6">CONLAT_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">O'</font> = <font color="Maroon">O</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2">Attributes</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> ;<br/>
<a NAME="E3:28"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">O'</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7">CONLAT_1:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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>set </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> ;<br/>
<a NAME="E4:28"/><i><font color="Green">E25</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:28_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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>then consider </b><font color="Maroon">a</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:28_1"/>
for <font color="Olive">o'</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o'</font> be   <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o'</font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E3:28_1_1"/>
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:28_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3">Information</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5">CONLAT_1:def 5</a></i>;<br/>
<a NAME="E5:28_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">o</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3">Information</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1_1"/>
<font color="Maroon">u'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">o'</font>
 <b>by </b><i>, <a class="ref" href="conlat_1.html#D5">CONLAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:28"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:28_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Olive">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="conlat_1.html#NM2">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span><b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; ( for <font color="Olive">o</font> being  <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O'</font> holds <br/><font color="Maroon">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">o</font> ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:28_2"/>
for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o'</font> be   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:28_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">o'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o'</font> as   <a href="conlat_1.html#NM3">Attribute</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span> ;<br/>
<a NAME="E3:28_2_1"/>
<font color="Maroon">a</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">o</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:28_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">o</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3">Information</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5">CONLAT_1:def 5</a></i>;<br/>
<a NAME="E5:28_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">o'</font>,<font color="Maroon">u'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3">Information</a> of <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:28_2_1"/>
<font color="Maroon">o'</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">u'</font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D5">CONLAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:28_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font> : for <font color="Olive">o</font> being  <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2">AttributeDerivation</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="conlat_2.html#K7" target="_self">.:</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">O</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
