<?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="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">a</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> : <font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Olive">a'</font> </span>}</span> 
 ;<br/><a NAME="E3:18_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="conlat_1.html#T1">CONLAT_1:1</a></i>;<br/><a NAME="E4:18_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <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="E6:18_1"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</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>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) &amp; <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> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <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="E8:18_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) &amp; <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> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <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> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E9:18_1"/><i><font color="Green">E26</font></i>: 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 <b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/><a NAME="E10:18_1"/>
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">O'</font>
 <b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/><a NAME="E11:18_1"/><b>then </b>
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="conlat_1.html#T4">CONLAT_1:4</a></i>;<br/><a NAME="E12:18_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> 
 <b>by </b><i><a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/><b>hence </b><a NAME="E13:18_1"/>
<span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</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#T47">CONLAT_1:47</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:18"/>
<span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="conlat_1.html#K12">@</a>  <a href="conlat_1.html#R2">is-SubConcept-of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#T47">CONLAT_1:47</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">o</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D19">CONLAT_1:def 19</a></i>;<br/>
<b>consider </b><font color="Maroon">O</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <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="E6:18"/><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K4" target="_self">gamma</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>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O</font>,<font color="Maroon">A</font> #) &amp; <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> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">O'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1">Objects</a> of <font color="Maroon">C</font>, <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="E8:18"/><i><font color="Green">E41</font></i>: 
( <span class="p1">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="conlat_1.html#G3">ConceptStr</a>(# <font color="Maroon">O'</font>,<font color="Maroon">A'</font> #) &amp; <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> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <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> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E9:18"/>
the <a href="conlat_1.html#U4">Extent</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_2.html#K5" target="_self">delta</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="conlat_1.html#K12">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">O'</font>
 
<b>by </b><i>, <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O'</font>
 
<b>by </b><i>, , <a class="ref" href="conlat_1.html#D24">CONLAT_1:def 24</a></i>;<br/>
<b>set </b><font color="Maroon">aa</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">o'</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">oo</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"><span class="p2">(<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> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E11:18"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <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> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<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> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="conlat_1.html#T11">CONLAT_1:11</a></i>;<br/>
<a NAME="E12:18"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#T7">CONLAT_1:7</a></i>;<br/>
<a NAME="E13:18"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E14:18"/><b>then </b>
<font color="Maroon">a</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> : <font color="Maroon">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#T1">CONLAT_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E16:18"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">b</font> )
 ;<br/>
<b>thus </b><a NAME="E17:18"/>
<font color="Maroon">o</font> <a href="conlat_1.html#R1">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


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