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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b1">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <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> #)
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/>

<a NAME="E3:50"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &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_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D4" onmouseover="rs('filter_0/D4')" onmouseout="rh()">FILTER_0:def 4</a></span>;<br/>
<a NAME="E4:50"/>
( <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c2">L2</font> &amp; <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> is   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="filter_2.html#T15" target="_self" onmouseover="rs('filter_2/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:50"/>
( <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="filter_0.html#D4" onmouseover="rs('filter_0/D4')" onmouseout="rh()">FILTER_0:def 4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><br/>

<a NAME="E6:50"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
( <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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> &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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D9" target="_self" onmouseover="rs('filter_2/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E7:50"/>
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> is   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c2">L2</font> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> is   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="filter_2.html#T16" target="_self" onmouseover="rs('filter_2/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:50"/>
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="filter_2.html#D9" target="_self" onmouseover="rs('filter_2/D9')" onmouseout="rh()">Def9</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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