<?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">A1</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>, <font color="Maroon">A2</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font>;<br/>



<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E2:33"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_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" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<a NAME="E2:33_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> <font color="Maroon">A'</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
<b>by </b><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="E4:33_1"/><i><font color="Green">E32</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" target="_self">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="E6:33_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> <font color="Maroon">A1</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><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="E1:33_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:33_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33_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> <font color="Maroon">A1</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<a NAME="E8:33_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:33_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> <font color="Maroon">A2</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><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="E1:33_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_2"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:33_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> <font color="Maroon">A2</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<a NAME="E11:33_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<a NAME="E2:33_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:33_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <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">A1</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span>  &amp; <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">A2</font> holds <br/><font color="Olive">o</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:33_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x1</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">A1</font> holds <br/><font color="Maroon">x1</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">x2</font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E7:33_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x2</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">A2</font> holds <br/><font color="Maroon">x2</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</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/>;<br/>
<a NAME="E9:33_2"/>
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">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> holds <br/><font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><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="E1:33_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_2_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_2_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_2_1"/>
<font color="Maroon">x</font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:33_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" target="_self">is-connected-with</a> <font color="Olive">a</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E11:33_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" target="_self">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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
