<?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/>

<a NAME="E1:31"/><i><font color="Green">E29</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2">LATTICE3:def 2</a></i>;<br/>
<a NAME="E2:31"/><i><font color="Green">E32</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2">LATTICE3:def 2</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span><br/> for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<a NAME="E6:31_1"/>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E7:31_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E8:31_1"/><b>then </b><i><font color="Green">E36</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <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</font> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <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</font> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E11:31_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E12:31_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</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="E2:31_1_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <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</font> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</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> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E16:31_1"/>
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:31_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">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" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E18:31_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">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" target="_self">AttributeDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E20:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E21:31_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</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> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E25:31_1"/><i><font color="Green">E43</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6" target="_self">Def6</a>, <a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E28:31_1"/>
<font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/>
<a NAME="E29:31_1"/><b>then </b>
<font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E30:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E31:31_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E32:31_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span><br/> for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>  st <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:31_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:31_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<a NAME="E6:31_2"/>
<font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b>
<font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E8:31_2"/><b>then </b><i><font color="Green">E45</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</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> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</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> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E11:31_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E12:31_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_2_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</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="E2:31_2_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <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</font> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<a NAME="E15:31_2"/>
<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">x'</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</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:31_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<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"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <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</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E17:31_2"/>
<font color="Maroon">x</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</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" target="_self">ObjectDerivation</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E18:31_2"/><b>then </b>
<font color="Maroon">x</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</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:31_2"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E20:31_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U2" target="_self">Attributes</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <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</font> <b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <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</font> ;<br/>
<a NAME="E25:31_2"/><i><font color="Green">E51</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D6" target="_self">Def6</a>, <a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E28:31_2"/>
<font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>;<br/>
<a NAME="E29:31_2"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E30:31_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice3.html#K2">LattRel</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<a NAME="E31:31_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="conlat_1.html#U1" target="_self">Objects</a> of <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="conlat_1.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E32:31_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:31"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K3" target="_self">phi</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="conlat_1.html#K4" target="_self">psi</a> <font color="Maroon">C</font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="conlat_1.html#V5" target="_self">co-Galois</a>
 <b>by </b><i>, </i>;<br/>


</div>
