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

<b>consider </b><font color="Maroon" title="c2">o</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c3">O</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">A</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E2:43_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">t</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E4:43_1_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">o'</font> where <font color="Olive" title="b1">o'</font> is   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> holds <br/><font color="Olive" title="b1">o'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b2">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D7" target="_self" title="CONLAT_1:def.7">Def7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x'</font> being   <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:43_1_1_2"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c7">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> holds <br/><font color="Maroon" title="c7">x'</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E5:43_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E4:43_1_1_2"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">O</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U1" title="CONLAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E6:43_1_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:43_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font> : for <font color="Olive" title="b2">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">t</font> holds <br/><font color="Olive" title="b2">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Olive" title="b1">a</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6" target="_self" title="CONLAT_1:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x'</font> being   <a href="conlat_1.html#NM3" title="CONLAT_1:NM.3">Attribute</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:43_1_1_3"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="conlat_1.html#NM2" title="CONLAT_1:NM.2">Object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">t</font> holds <br/><font color="Olive" title="b1">o</font> <a href="conlat_1.html#R1" title="CONLAT_1:pred.1">is-connected-with</a> <font color="Maroon" title="c8">x'</font> ) )
 ;<br/>
<b>thus </b><a NAME="E5:43_1_1_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="conlat_1.html#E4:43_1_1_3"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="conlat_1.html#U2" title="CONLAT_1:sel.2">Attributes</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<b>set </b><font color="Maroon" title="c8">M'</font> =  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c7">A</font> #);<br/>
<a NAME="E8:43_1_1"/>
not  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c7">A</font> #) is <a href="conlat_1.html#V7" title="CONLAT_1:attr.7">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_1_4"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:43_1_1_4"/>
<font color="Maroon" title="c5">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#T5" target="_self" title="CONLAT_1:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_4"/>
not  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c7">A</font> #) is <a href="conlat_1.html#V7" title="CONLAT_1:attr.7">empty</a>
 <b>by </b><i><a class="txt" href="conlat_1.html#E1:43_1_1_4"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="conlat_1.html#D11" target="_self" title="CONLAT_1:def.11">Def11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">M'</font> =  <a href="conlat_1.html#G3" title="CONLAT_1:aggr.3">ConceptStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c7">A</font> #) as  non <a href="conlat_1.html#V7" title="CONLAT_1:attr.7">empty</a>  <a href="conlat_1.html#L3" title="CONLAT_1:struct.3">ConceptStr</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> the <a href="conlat_1.html#U4" title="CONLAT_1:sel.4">Extent</a> of <font color="Maroon" title="c9">M'</font> = 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font>
<b>by </b><i><a class="ref" href="conlat_1.html#T7" target="_self" title="CONLAT_1:th.7">Th7</a></i>
<br/>.= 
the <a href="conlat_1.html#U5" title="CONLAT_1:sel.5">Intent</a> of <font color="Maroon" title="c9">M'</font>

;<br/>
<a NAME="E11:43_1_1"/><b>then </b>
<font color="Maroon" title="c9">M'</font> is <a href="conlat_1.html#V9" title="CONLAT_1:attr.9">concept-like</a>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D13" target="_self" title="CONLAT_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E12:43_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="conlat_1.html#L3" title="CONLAT_1:struct.3">ConceptStr</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="conlat_1.html#V9" title="CONLAT_1:attr.9">concept-like</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="conlat_1.html#V7" title="CONLAT_1:attr.7">empty</a> )
 ;<br/>


</div>
