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<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>, <font color="Olive" title="b2">E1</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="b3">D2</font>, <font color="Olive" title="b4">E2</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="b3">D2</font> &amp; <font color="Olive" title="b2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">E2</font> holds <br/><font color="Olive" title="b1">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b2">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b4">E2</font></span><br/><b>let </b><font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">E1</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>, <font color="Olive" title="b2">E2</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> &amp; <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E2</font> holds <br/><font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b2">E2</font></span><br/><b>let </b><font color="Maroon" title="c5">D2</font>, <font color="Maroon" title="c6">E2</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="c5">D2</font> &amp; <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E2</font> implies <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font> )</span><br/>





<b>assume </b><a NAME="E1:68"/><i><font color="Green" title="E26">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="c5">D2</font> &amp; <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font></span><br/>

<b>thus </b><a NAME="E2:68"/>
<font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font>
  <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"> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font> )</span><br/>

<b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">E1</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a'</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E5:68_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:68_1"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:68_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:68_1"><i><font color="Green" title="E27">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font> )</span><br/>

<b>assume </b><a NAME="E3:68"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c6">E2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E5:68"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E2</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a'</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c11">b'</font> = <font color="Maroon" title="c9">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E7:68"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">b'</font>
 
<b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:68"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c4">E1</font>
 <b>by </b><i><a class="txt" href="#E1:68"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E27">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
