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<span class="kw">let </span><font color="Maroon" title="c1">L</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>, <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="c1">L</font> holds  <font color="Olive" title="b1">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Olive" title="b2">D2</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b2">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Olive" title="b1">D1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">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="c1">L</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"> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D1" target="_self" onmouseover="rs('filter_2/D1')" onmouseout="rh()">FILTER_2:def 1</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"> ( <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">D2</font> iff <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:63"/>
( <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">D2</font> implies <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</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"> ( <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</font> implies <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">D2</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1"/>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</font></span><br/>

<a NAME="E2:63_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:63_1"/>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:63"/>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c2">D1</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"> <font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">D2</font></span><br/>

<a NAME="E3:63"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:63"/>
<font color="Maroon" title="c4">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K8" title="FILTER_2:func.8">"\/"</a> <font color="Maroon" title="c3">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"> verum</span><br/>


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