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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="conlat_1.html#NM1">FormalContext</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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<sub>1</sub></font>;<br/>





<a NAME="E1:23"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U3" target="_self">Information</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="conlat_1.html#NM3">Attribute</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>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</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="E3:23_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="conlat_1.html#E5:23_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="conlat_1.html#E5:23_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E8:23_1"/>
<font color="Maroon">c<sub>7</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="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="conlat_1.html#NM3">Attribute</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>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="conlat_1.html#E2:23_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conlat_1.html#E5:23_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:23_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="conlat_1.html#NM3">Attribute</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>3</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<a NAME="E11:23_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E5:23_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conlat_1.html#E10:23_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:23_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="conlat_1.html#U3" target="_self">Information</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E5:23_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="conlat_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:23"/>
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U3" target="_self">Information</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U3" target="_self">Information</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</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="E2:23_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</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>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:23_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="conlat_1.html#NM3">Attribute</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>3</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="conlat_1.html#NM3">Attribute</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:23_2_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E4:23_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="conlat_1.html#E2:23_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="conlat_1.html#E1:23_2_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="conlat_1.html#E3:23_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:23_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:23_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="conlat_1.html#E3:23_2_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="conlat_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:23_2"/><b>then </b>
<font color="Maroon">c<sub>5</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="conlat_1.html#NM2">Object</a> of <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="conlat_1.html#NM3">Attribute</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>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="conlat_1.html#R1" target="_self">is-connected-with</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:23_2"/>
<font color="Maroon">c<sub>4</sub></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<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" target="_self">AttributeDerivation</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> iff <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> the <a href="conlat_1.html#U3" target="_self">Information</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:23"><i><font color="Green">E13</font></i></a></i>;<br/>


</div>
