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

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D1</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font><br/> for <font color="Olive" title="b2">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L2</font>  st  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Olive" title="b1">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">D2</font> holds <br/>( <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/><b>let </b><font color="Maroon" title="c3">D1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L2</font>  st  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D2</font> holds <br/>( <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Olive" title="b1">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/><b>let </b><font color="Maroon" title="c4">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> implies ( <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> ) )</span><br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E21">A1</font></i>: 
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L1</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L2</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L2</font> #) &amp; <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/>

<a NAME="E2:56"/><b>then </b>
( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; ( for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/><span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> ) &amp; <font color="Maroon" title="c4">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; ( for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c4">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/><span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> ) &amp; <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> is    <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c2">L2</font> &amp; <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> is    <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c1">L1</font> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T16" target="_self" title="FILTER_2:th.16">Th16</a>, <a class="ref" href="filter_0.html#D5" title="FILTER_0:def.5">FILTER_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:56"/>
( <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> )
 <b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E21">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span><br/>

<a NAME="E4:56"/>
( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; ( for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/><span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> ) &amp; <font color="Maroon" title="c4">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; ( for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c4">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/><span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> ) &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> is    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c2">L2</font> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> is    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L1</font> )
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="filter_2.html#D11" target="_self" title="FILTER_2:def.11">Def11</a>, <a class="ref" href="filter_2.html#T17" target="_self" title="FILTER_2:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E5:56"/>
( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E21">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
