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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#V11" title="LATTICES:attr.11">distributive</a>   <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>  <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"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  iff  for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  holds <br/><font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:56"/>
( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  implies  for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  holds <br/><font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</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">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  holds <br/><font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  ) implies <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E22">YY</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</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">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  holds <br/><font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</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">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  implies <font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E23">Y0</font></span>: 
( <font color="Maroon" title="c2">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</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="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> </span><br/>

<a NAME="E3:56_1"/>
 for <font color="Olive" title="b1">G</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</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="c3">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1_1"/><span class="lab"><font color="Green" title="E24">y1</font></span>: 
( <font color="Maroon" title="c3">G</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">G</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">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font></span><br/>

<a NAME="E2:56_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">y3</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</a></span>;<br/>
<a NAME="E3:56_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="filter_2.html#K5" title="FILTER_2:func.5">(.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_2.html#K5" title="FILTER_2:func.5">.&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1"><span class="lab"><font color="Green" title="E24">y1</font></span></a></span>;<br/>
<a NAME="E4:56_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> is  <a href="filter_2.html#V1" title="FILTER_2:attr.1">max-ideal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E23">Y0</font></span></a>, <a class="ref" href="filter_2.html#T57" onmouseover="rs('filter_2/T57')" onmouseout="rh()">FILTER_2:57</a>, <a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E22">YY</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56_1_1"/>
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D8" onmouseover="rs('filter_2/D8')" onmouseout="rh()">FILTER_2:def 8</a>, <a class="txt" href="#E1:56_1_1"><span class="lab"><font color="Green" title="E24">y1</font></span></a>, <a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E25">y3</font></span></a></span>; <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">hence </span><a NAME="E4:56_1"/>
<font color="Maroon" title="c2">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E23">Y0</font></span></a></span>; <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="E2:56"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 for <font color="Olive" title="b1">I</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">I</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  holds <br/><font color="Olive" title="b1">I</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</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="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:56"/>
 not <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</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"> contradiction</span><br/>

<a NAME="E4:56"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V16" title="LATTICES:attr.16">complemented</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:56"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds  not <font color="Olive" title="b1">b</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">a</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">I0</font> =  <a href="latticea.html#K6" title="LATTICEA:func.6">PseudoComplements</a> <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">I0</font> =  <a href="latticea.html#K6" title="LATTICEA:func.6">PseudoComplements</a> <font color="Maroon" title="c2">a</font> as   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E8:56"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E9:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">C2</font></span>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font>
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> ) </span> } </span> ;<br/>
<a NAME="E10:56"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> ) </span> } </span>  or <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_2"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> ) </span> } </span> 
 ; <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="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:56_2"/><span class="lab"><font color="Green" title="E25">C1</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp;  ex <font color="Olive" title="b1">y</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="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c7">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">y</font> ) )
 ;<br/>
<span class="kw">thus </span><a NAME="E4:56_2"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2"><span class="lab"><font color="Green" title="E25">C1</font></span></a></span>; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">I1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">y</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="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</font> ) </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E12:56"/>
 for <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 <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> holds <br/><font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c7">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> implies <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_3"/><span class="lab"><font color="Green" title="E25">C0</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</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="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:56_3"/><span class="lab"><font color="Green" title="E26">C1</font></span>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> &amp;  ex <font color="Olive" title="b1">y</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="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">y</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:56_3"/><span class="lab"><font color="Green" title="E27">CC</font></span>: 
( <font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E26">C1</font></span></a></span>;<br/>
<a NAME="E6:56_3"/>
<font color="Maroon" title="c7">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E26">C1</font></span></a>, <a class="txt" href="#E1:56_3"><span class="lab"><font color="Green" title="E25">C0</font></span></a>, <a class="ref" href="lattices.html#T7" onmouseover="rs('lattices/T7')" onmouseout="rh()">LATTICES:7</a>, <a class="txt" href="#E5:56_3"><span class="lab"><font color="Green" title="E27">CC</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56_3"/>
<font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_3"><span class="lab"><font color="Green" title="E27">CC</font></span></a></span>; <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>
<a NAME="E13:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">C1</font></span>: 
<font color="Maroon" title="c6">I1</font> is  <a href="lattices.html#V18" title="LATTICES:attr.18">initial</a> 
 
;<br/>
<a NAME="E14:56"/>
 for <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 <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> &amp; <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> implies <font color="Maroon" title="c7">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_4"/><span class="lab"><font color="Green" title="E26">d0</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font> &amp; <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</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="c7">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:56_4"/><span class="lab"><font color="Green" title="E27">d1</font></span>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp;  ex <font color="Olive" title="b1">y</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="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">y</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:56_4"/><span class="lab"><font color="Green" title="E28">e1</font></span>: 
( <font color="Maroon" title="c10">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c9">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_4"><span class="lab"><font color="Green" title="E27">d1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:56_4"/><span class="lab"><font color="Green" title="E29">d2</font></span>: 
( <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> &amp;  ex <font color="Olive" title="b1">y</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="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_4"><span class="lab"><font color="Green" title="E26">d0</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:56_4"/><span class="lab"><font color="Green" title="E30">e2</font></span>: 
( <font color="Maroon" title="c12">y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_4"><span class="lab"><font color="Green" title="E29">d2</font></span></a></span>;<br/>
<a NAME="E10:56_4"/><span class="lab"><font color="Green" title="E31">HH</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D25" onmouseover="rs('lattices/D25')" onmouseout="rh()">LATTICES:def 25</a>, <a class="txt" href="#E5:56_4"><span class="lab"><font color="Green" title="E28">e1</font></span></a>, <a class="txt" href="#E9:56_4"><span class="lab"><font color="Green" title="E30">e2</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E32">FF</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" onmouseover="rs('lattices/D5')" onmouseout="rh()">LATTICES:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" onmouseover="rs('lattices/D5')" onmouseout="rh()">LATTICES:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D5" onmouseover="rs('lattices/D5')" onmouseout="rh()">LATTICES:def 5</a></span>
;<br/>
<a NAME="E12:56_4"/>
<font color="Maroon" title="c9">x1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">x2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T4" onmouseover="rs('filter_0/T4')" onmouseout="rh()">FILTER_0:4</a>, <a class="txt" href="#E5:56_4"><span class="lab"><font color="Green" title="E28">e1</font></span></a>, <a class="txt" href="#E9:56_4"><span class="lab"><font color="Green" title="E30">e2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:56_4"/>
<font color="Maroon" title="c7">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_4"><span class="lab"><font color="Green" title="E31">HH</font></span></a>, <a class="txt" href="#E3:56_4"><span class="lab"><font color="Green" title="E27">d1</font></span></a>, <a class="txt" href="#E7:56_4"><span class="lab"><font color="Green" title="E29">d2</font></span></a>, <a class="txt" href="#E11:56_4"><span class="lab"><font color="Green" title="E32">FF</font></span></a></span>; <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>
<a NAME="E15:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">CC</font></span>: 
<font color="Maroon" title="c6">I1</font> is  <a href="lattices.html#V21" title="LATTICES:attr.21">join-closed</a> 
 
;<br/>
<a NAME="E16:56"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
;<br/>
<a NAME="E17:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">ZZ</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56"><span class="lab"><font color="Green" title="E24">C2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">I1</font> = <font color="Maroon" title="c6">I1</font> as   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E25">C1</font></span></a>, <a class="txt" href="#E15:56"><span class="lab"><font color="Green" title="E26">CC</font></span></a></span>;<br/>
<a NAME="E19:56"/><span class="lab"><font color="Green" title="E28">q1</font></span>: 
<font color="Maroon" title="c4">I0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">I1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I0</font> or <font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_5"/><span class="lab"><font color="Green" title="E29">C0</font></span>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I0</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="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:56_5"/><span class="lab"><font color="Green" title="E30">C1</font></span>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 ;<br/>
<a NAME="E4:56_5"/>
<font color="Maroon" title="c9">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T5" onmouseover="rs('lattices/T5')" onmouseout="rh()">LATTICES:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56_5"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">I1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_5"><span class="lab"><font color="Green" title="E30">C1</font></span></a>, <a class="txt" href="#E1:56_5"><span class="lab"><font color="Green" title="E29">C0</font></span></a></span>; <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>
<a NAME="E20:56"/><span class="lab"><font color="Green" title="E29">J1</font></span>: 
 not  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">I1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_6"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">I1</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:56_6"/><span class="lab"><font color="Green" title="E30">D1</font></span>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp;  ex <font color="Olive" title="b1">y</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="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp; <font color="Maroon" title="c8">x1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">y</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:56_6"/><span class="lab"><font color="Green" title="E31">D2</font></span>: 
( <font color="Maroon" title="c9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_6"><span class="lab"><font color="Green" title="E30">D1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:56_6"/><span class="lab"><font color="Green" title="E32">C1</font></span>: 
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_6"><span class="lab"><font color="Green" title="E31">D2</font></span></a></span>;<br/>
<a NAME="E8:56_6"/>
<font color="Maroon" title="c9">y</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_6"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="txt" href="#E5:56_6"><span class="lab"><font color="Green" title="E31">D2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:56_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E23">A2</font></span></a></span>; <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">set </span><font color="Maroon" title="c8">FF</font> = <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>;<br/>
<a NAME="E21:56"/>
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T65" onmouseover="rs('filter_2/T65')" onmouseout="rh()">FILTER_2:65</a></span>;<br/>
<a NAME="E22:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">J2</font></span>: 
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T64" onmouseover="rs('filter_2/T64')" onmouseout="rh()">FILTER_2:64</a></span>;<br/>
<a NAME="E23:56"/><span class="kw">then </span>
<span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c7">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56"><span class="lab"><font color="Green" title="E29">J1</font></span></a>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">J0</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E25:56"/><span class="lab"><font color="Green" title="E31">B1</font></span>: 
( <font color="Maroon" title="c9">J0</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c7">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">J0</font> &amp; <font color="Maroon" title="c9">J0</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T19" target="_self" onmouseover="rs('latticea/T19')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E26:56"/><span class="lab"><font color="Green" title="E32">S4</font></span>: 
<font color="Maroon" title="c9">J0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:56"><span class="lab"><font color="Green" title="E31">B1</font></span></a>, <a class="ref" href="zfmisc_1.html#T48" onmouseover="rs('zfmisc_1/T48')" onmouseout="rh()">ZFMISC_1:48</a>, <a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E30">J2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">T</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">D</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">J0</font> as   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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E32">S4</font></span></a>, <a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>;<br/>
<a NAME="E28:56"/>
 for <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 <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.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="c11">D</font> holds <br/><font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c12">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> implies <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_7"/><span class="lab"><font color="Green" title="E33">k0</font></span>: 
( <font color="Maroon" title="c12">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q</font> &amp; <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</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="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font></span><br/>

<a NAME="E2:56_7"/><span class="kw">then </span>
( <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  not <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:56_7"/><span class="kw">then </span>
 not <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_7"><span class="lab"><font color="Green" title="E33">k0</font></span></a>, <a class="ref" href="filter_2.html#T21" onmouseover="rs('filter_2/T21')" onmouseout="rh()">FILTER_2:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_7"/>
<font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <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>
<a NAME="E29:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">j1</font></span>: 
<font color="Maroon" title="c11">D</font> is  <a href="lattices.html#V19" title="LATTICES:attr.19">final</a> 
 
;<br/>
<a NAME="E30:56"/>
 for <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 <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> implies <font color="Maroon" title="c12">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_8"/>
( <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</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="c12">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font></span><br/>

<a NAME="E2:56_8"/><span class="kw">then </span>
( <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  not <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font> &amp; <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  not <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:56_8"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D10" onmouseover="rs('filter_2/D10')" onmouseout="rh()">FILTER_2:def 10</a>, <a class="txt" href="#E25:56"><span class="lab"><font color="Green" title="E31">B1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_8"/>
<font color="Maroon" title="c12">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <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>
<a NAME="E31:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">zc</font></span>: 
<font color="Maroon" title="c11">D</font> is  <a href="lattices.html#V20" title="LATTICES:attr.20">meet-closed</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">F</font> = <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">D</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> as   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E33:56"/>
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
;<br/>
<a NAME="E34:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">H1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E35:56"/><span class="lab"><font color="Green" title="E36">h2</font></span>: 
<font color="Maroon" title="c11">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">F</font>
 
<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="E36:56"/><span class="lab"><font color="Green" title="E37">G1</font></span>: 
 not  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">J0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:56"><span class="lab"><font color="Green" title="E31">B1</font></span></a>, <a class="ref" href="zfmisc_1.html#T48" onmouseover="rs('zfmisc_1/T48')" onmouseout="rh()">ZFMISC_1:48</a>, <a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E30">J2</font></span></a></span>;<br/>
<a NAME="E37:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">G2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">J0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a></span>;<br/>
<a NAME="E38:56"/>
<font color="Maroon" title="c11">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E39:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">mm</font></span>: 
<font color="Maroon" title="c11">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E36">h2</font></span></a></span>;<br/>
<a NAME="E40:56"/>
<font color="Maroon" title="c12">F</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c4">I0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_9"/>
<font color="Maroon" title="c12">F</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c4">I0</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:56_9"/><span class="lab"><font color="Green" title="E40">H1</font></span>: 
( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E5:56_9"/><span class="lab"><font color="Green" title="E41">C1</font></span>: 
( <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_9"><span class="lab"><font color="Green" title="E40">H1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:56_9"/><span class="lab"><font color="Green" title="E42">n1</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c9">J0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a>, <a class="txt" href="#E37:56"><span class="lab"><font color="Green" title="E38">G2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c15">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_9"><span class="lab"><font color="Green" title="E42">n1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">bb</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E10:56_9"/><span class="lab"><font color="Green" title="E43">m2</font></span>: 
( <font color="Maroon" title="c17">bb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c17">bb</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T3" onmouseover="rs('lattice4/T3')" onmouseout="rh()">LATTICE4:3</a>, <a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E34">zc</font></span></a>, <a class="txt" href="#E5:56_9"><span class="lab"><font color="Green" title="E41">C1</font></span></a>, <a class="txt" href="#E3:56_9"><span class="lab"><font color="Green" title="E40">H1</font></span></a>, <a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E33">j1</font></span></a></span>;<br/>
<a NAME="E11:56_9"/><span class="lab"><font color="Green" title="E44">W1</font></span>: 
( <font color="Maroon" title="c17">bb</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">x1</font> &amp;  not <font color="Maroon" title="c17">bb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">J0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_9"><span class="lab"><font color="Green" title="E43">m2</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E12:56_9"/>
<font color="Maroon" title="c17">bb</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T6" onmouseover="rs('lattices/T6')" onmouseout="rh()">LATTICES:6</a></span>;<br/>
<a NAME="E13:56_9"/><span class="kw">then </span>
<font color="Maroon" title="c17">bb</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_9"><span class="lab"><font color="Green" title="E41">C1</font></span></a>, <a class="ref" href="boolealg.html#T9" onmouseover="rs('boolealg/T9')" onmouseout="rh()">BOOLEALG:9</a>, <a class="txt" href="#E10:56_9"><span class="lab"><font color="Green" title="E43">m2</font></span></a>, <a class="ref" href="filter_0.html#T7" onmouseover="rs('filter_0/T7')" onmouseout="rh()">FILTER_0:7</a></span>;<br/>
<a NAME="E14:56_9"/><span class="kw">then </span>
<font color="Maroon" title="c17">bb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">I0</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E15:56_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56_9"><span class="lab"><font color="Green" title="E44">W1</font></span></a>, <a class="txt" href="#E25:56"><span class="lab"><font color="Green" title="E31">B1</font></span></a>, <a class="txt" href="#E19:56"><span class="lab"><font color="Green" title="E28">q1</font></span></a></span>; <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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">J1</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E42:56"/><span class="lab"><font color="Green" title="E40">B2</font></span>: 
( <font color="Maroon" title="c13">J1</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c4">I0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">J1</font> &amp; <font color="Maroon" title="c13">J1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c12">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T19" target="_self" onmouseover="rs('latticea/T19')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E43:56"/>
<font color="Maroon" title="c13">J1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E35">H1</font></span></a>, <a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E36">h2</font></span></a>, <a class="txt" href="#E42:56"><span class="lab"><font color="Green" title="E40">B2</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<a NAME="E44:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">S1</font></span>: 
<font color="Maroon" title="c13">J1</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</a></span>;<br/>
<a NAME="E45:56"/><span class="lab"><font color="Green" title="E42">S2</font></span>: 
<font color="Maroon" title="c9">J0</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E37">G1</font></span></a>, <a class="ref" href="subset_1.html#D6" onmouseover="rs('subset_1/D6')" onmouseout="rh()">SUBSET_1:def 6</a></span>;<br/>
<a NAME="E46:56"/><span class="lab"><font color="Green" title="E43">S3</font></span>: 
<font color="Maroon" title="c13">J1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">J0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:56"><span class="lab"><font color="Green" title="E31">B1</font></span></a>, <a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E27">ZZ</font></span></a>, <a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E36">h2</font></span></a>, <a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E35">H1</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="txt" href="#E42:56"><span class="lab"><font color="Green" title="E40">B2</font></span></a></span>;<br/>
<a NAME="E47:56"/>
<font color="Maroon" title="c13">J1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">J0</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> or <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">J0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_10"/><span class="lab"><font color="Green" title="E44">f3</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</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="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">J0</font></span><br/>

<a NAME="E2:56_10"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:56"><span class="lab"><font color="Green" title="E39">mm</font></span></a>, <a class="txt" href="#E42:56"><span class="lab"><font color="Green" title="E40">B2</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_10"/>
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">J0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_10"><span class="lab"><font color="Green" title="E44">f3</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <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>
<a NAME="E48:56"/><span class="kw">then </span>
 not <font color="Maroon" title="c13">J1</font> is  <a href="latticea.html#V3" title="LATTICEA:attr.3">maximal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:56"><span class="lab"><font color="Green" title="E42">S2</font></span></a>, <a class="txt" href="#E46:56"><span class="lab"><font color="Green" title="E43">S3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E49:56"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:56"><span class="lab"><font color="Green" title="E40">B2</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E44:56"><span class="lab"><font color="Green" title="E41">S1</font></span></a></span>; <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>
