<?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  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:58"/>
( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  implies  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</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"> (  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</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="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_1"/>
<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">  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">LL</font> = <font color="Maroon" title="c1">L</font> as   <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a> ;<br/>
<span class="kw">assume </span><a NAME="E3:58_1"/>
 not  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E5:58_1"/><span class="lab"><font color="Green" title="E23">F1</font></span>: 
( <font color="Maroon" title="c3">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">P1</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="E7:58_1"/><span class="lab"><font color="Green" title="E24">B1</font></span>: 
( <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">P1</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c5">P1</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="#E5:58_1"><span class="lab"><font color="Green" title="E23">F1</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Q1</font> being   <a href="filter_2.html#NM1" title="FILTER_2:NM.1">Ideal</a> of <font color="Maroon" title="c2">LL</font><span class="kw"> such that </span><br/><a NAME="E9:58_1"/><span class="lab"><font color="Green" title="E25">b1</font></span>: 
( <font color="Maroon" title="c6">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c6">Q1</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c6">Q1</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="#E5:58_1"><span class="lab"><font color="Green" title="E23">F1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E27">f1</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Q</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/><a NAME="E2:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E23">F1</font></span></a></span>;<br/><a NAME="E3:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">P</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#T105" onmouseover="rs('xboole_1/T105')" onmouseout="rh()">XBOOLE_1:105</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">P</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></span>;<br/><a NAME="E6:58_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font> &amp;  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</font></span></a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a>, <a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E28">A4</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">LL</font> ;<br/><a NAME="E8:58_1_1"/>
<font color="Maroon" title="c6">Q1</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="c2">LL</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="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</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="E9:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">Q1</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="ref" href="filter_2.html#T57" onmouseover="rs('filter_2/T57')" onmouseout="rh()">FILTER_2:57</a>, <a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</font></span></a></span>;<br/><a NAME="E10:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B2</font></span>: 
 not <font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E27">f1</font></span></a>, <a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</font></span></a>, <a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="filter_2.html#T58" onmouseover="rs('filter_2/T58')" onmouseout="rh()">FILTER_2:58</a></span>;<br/><a NAME="E11:58_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></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="c2">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T20" onmouseover="rs('lattices/T20')" onmouseout="rh()">LATTICES:20</a></span>;<br/><a NAME="E12:58_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T24" onmouseover="rs('filter_2/T24')" onmouseout="rh()">FILTER_2:24</a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:58_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a>, <a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E10:58_1_1"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="ref" href="filter_2.html#D10" onmouseover="rs('filter_2/D10')" onmouseout="rh()">FILTER_2:def 10</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1_2"><span class="kw">now </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">  not <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_2"/><span class="lab"><font color="Green" title="E27">f1</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</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/><a NAME="E2:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">Q</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E23">F1</font></span></a></span>;<br/><a NAME="E3:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">Q</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#T105" onmouseover="rs('xboole_1/T105')" onmouseout="rh()">XBOOLE_1:105</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:58_1_2"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">Q</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></span>;<br/><a NAME="E6:58_1_2"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a>, <a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</font></span></a>, <a class="txt" href="#E5:58_1_2"><span class="lab"><font color="Green" title="E28">A4</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">LL</font> ;<br/><a NAME="E8:58_1_2"/>
<font color="Maroon" title="c5">P1</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="c2">LL</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="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</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="E9:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">P1</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="ref" href="filter_2.html#T57" onmouseover="rs('filter_2/T57')" onmouseout="rh()">FILTER_2:57</a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a></span>;<br/><a NAME="E10:58_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">B2</font></span>: 
 not <font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_2"><span class="lab"><font color="Green" title="E27">f1</font></span></a>, <a class="txt" href="#E7:58_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a>, <a class="txt" href="#E5:58_1_2"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="filter_2.html#T58" onmouseover="rs('filter_2/T58')" onmouseout="rh()">FILTER_2:58</a></span>;<br/><a NAME="E11:58_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></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="c2">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T20" onmouseover="rs('lattices/T20')" onmouseout="rh()">LATTICES:20</a></span>;<br/><a NAME="E12:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T24" onmouseover="rs('filter_2/T24')" onmouseout="rh()">FILTER_2:24</a></span>;<br/><span class="kw">hence </span><a NAME="E13:58_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_2"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E10:58_1_2"><span class="lab"><font color="Green" title="E30">B2</font></span></a>, <a class="ref" href="filter_2.html#D10" onmouseover="rs('filter_2/D10')" onmouseout="rh()">FILTER_2:def 10</a>, <a class="txt" href="#E9:58_1"><span class="lab"><font color="Green" title="E25">b1</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="E12:58_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:58_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E23">F1</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="E2:58"/><span class="lab"><font color="Green" title="E23">d1</font></span>: 
 <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> is  <a href="latticea.html#V1" title="LATTICEA:attr.1">unordered</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="E3:58"/>
 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="E4:58"/><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="E6:58"/><span class="lab"><font color="Green" title="E24">C2</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">D</font> =  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">D</font> =  <a href="latticea.html#K7" title="LATTICEA:func.7">PseudoCocomplements</a> <font color="Maroon" title="c2">a</font> as   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D1</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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>;<br/>
<a NAME="E8:58"/><span class="lab"><font color="Green" title="E25">II</font></span>: 
<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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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">d</font>, <font color="Olive" title="b3">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="Olive" title="b2">d</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b2">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Olive" title="b3">q</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> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T35" onmouseover="rs('filter_0/T35')" onmouseout="rh()">FILTER_0:35</a></span>;<br/>
<a NAME="E9:58"/><span class="lab"><font color="Green" title="E26">AB</font></span>: 
<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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> or <font color="Maroon" title="c6">t</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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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"> <font color="Maroon" title="c6">t</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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">r</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="E3:58_2"/><span class="lab"><font color="Green" title="E27">I1</font></span>: 
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> &amp;  ex <font color="Olive" title="b1">d</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="Olive" title="b1">d</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Olive" title="b1">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Olive" title="b2">q</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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E25">II</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">q</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="E5:58_2"/><span class="lab"><font color="Green" title="E28">I2</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c8">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c9">q</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_2"><span class="lab"><font color="Green" title="E27">I1</font></span></a></span>;<br/>
<a NAME="E6:58_2"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_2"><span class="lab"><font color="Green" title="E28">I2</font></span></a>, <a class="ref" href="filter_0.html#T15" onmouseover="rs('filter_0/T15')" onmouseout="rh()">FILTER_0:15</a></span>;<br/>
<a NAME="E7:58_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">d</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="c8">d</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T5" onmouseover="rs('filter_0/T5')" onmouseout="rh()">FILTER_0:5</a></span>;<br/>
<a NAME="E8:58_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">d</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="c7">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_2"><span class="lab"><font color="Green" title="E28">I2</font></span></a>, <a class="ref" href="lattices.html#T7" onmouseover="rs('lattices/T7')" onmouseout="rh()">LATTICES:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:58_2"/>
<font color="Maroon" title="c6">t</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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_2"><span class="lab"><font color="Green" title="E27">I1</font></span></a>, <a class="txt" href="#E5:58_2"><span class="lab"><font color="Green" title="E28">I2</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="E10:58"/>
<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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">t</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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span>  or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_3"/>
<font color="Maroon" title="c6">t</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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></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="E3:58_3"/><span class="lab"><font color="Green" title="E27">a1</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t</font> &amp;  ex <font color="Olive" title="b1">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">x1</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">d</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="E5:58_3"/><span class="lab"><font color="Green" title="E28">a2</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_3"><span class="lab"><font color="Green" title="E27">a1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E6:58_3"/>
( <font color="Maroon" title="c8">d</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="c7">x1</font> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_3"><span class="lab"><font color="Green" title="E28">a2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:58_3"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_3"><span class="lab"><font color="Green" title="E27">a1</font></span></a>, <a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E25">II</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="E11:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">Z0</font></span>: 
<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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <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">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E26">AB</font></span></a></span>;<br/>
<a NAME="E12:58"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">D</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="E13:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">HH</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#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E27">Z0</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">D1</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"><font color="Maroon" title="c4">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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></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="E15:58"/><span class="lab"><font color="Green" title="E29">HJ</font></span>: 
 not  <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="c6">D1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_4"/>
 <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="c6">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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="E3:58_4"/><span class="lab"><font color="Green" title="E30">Z1</font></span>: 
( <font color="Maroon" title="c7">y</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> &amp;  ex <font color="Olive" title="b1">d</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">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E27">Z0</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">d</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="E5:58_4"/><span class="lab"><font color="Green" title="E31">Z2</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c8">d</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="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_4"><span class="lab"><font color="Green" title="E30">Z1</font></span></a></span>;<br/>
<a NAME="E6:58_4"/><span class="lab"><font color="Green" title="E32">z4</font></span>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">d</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c2">a</font>
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">x</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="E8:58_4"/><span class="lab"><font color="Green" title="E33">Z3</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">x</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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_4"><span class="lab"><font color="Green" title="E31">Z2</font></span></a></span>;<br/>
<a NAME="E9:58_4"/>
<font color="Maroon" title="c8">d</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="#E6:58_4"><span class="lab"><font color="Green" title="E32">z4</font></span></a>, <a class="txt" href="#E8:58_4"><span class="lab"><font color="Green" title="E33">Z3</font></span></a>, <a class="txt" href="#E3:58_4"><span class="lab"><font color="Green" title="E30">Z1</font></span></a>, <a class="txt" href="#E5:58_4"><span class="lab"><font color="Green" title="E31">Z2</font></span></a>, <a class="ref" href="lattices.html#T8" onmouseover="rs('lattices/T8')" onmouseout="rh()">LATTICES:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:58_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E24">C2</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">reconsider </span><font color="Maroon" title="c7">I0</font> = <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#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> 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="ref" href="filter_2.html#T25" onmouseover="rs('filter_2/T25')" onmouseout="rh()">FILTER_2:25</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">P</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="E18:58"/><span class="lab"><font color="Green" title="E30">W0</font></span>: 
( <font color="Maroon" title="c8">P</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c7">I0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c8">P</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c6">D1</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>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a>, <a class="txt" href="#E15:58"><span class="lab"><font color="Green" title="E29">HJ</font></span></a></span>;<br/>
<a NAME="E19:58"/>
<font color="Maroon" title="c8">P</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="#E18:58"><span class="lab"><font color="Green" title="E30">W0</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="#E13:58"><span class="lab"><font color="Green" title="E28">HH</font></span></a></span>;<br/>
<a NAME="E20:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">w0</font></span>: 
<font color="Maroon" title="c8">P</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/>
<span class="kw">set </span><font color="Maroon" title="c9">Pa</font> = <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Pa</font> = <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> 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="E22:58"/><span class="lab"><font color="Green" title="E32">ZZ</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_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span>
 
;<br/>
<a NAME="E23:58"/><span class="lab"><font color="Green" title="E33">ZL</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</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="E24:58"/><span class="lab"><font color="Green" title="E34">ZK</font></span>: 
<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</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="E25:58"/><span class="lab"><font color="Green" title="E35">ZM</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Pa</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#D9" onmouseover="rs('filter_2/D9')" onmouseout="rh()">FILTER_2:def 9</a></span>;<br/>
<a NAME="E26:58"/><span class="lab"><font color="Green" title="E36">kk</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">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="E27:58"/>
<font color="Maroon" title="c4">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="c6">D1</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="E28:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">hh</font></span>: 
<font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:58"><span class="lab"><font color="Green" title="E36">kk</font></span></a></span>;<br/>
<a NAME="E29:58"/><span class="lab"><font color="Green" title="E38">zx</font></span>: 
 not <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58"><span class="lab"><font color="Green" title="E28">HH</font></span></a>, <a class="txt" href="#E18:58"><span class="lab"><font color="Green" title="E30">W0</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="E30:58"/>
 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="c10">Pa</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_5"/><span class="lab"><font color="Green" title="E39">f2</font></span>: 
 <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="c10">Pa</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/>

<a NAME="E2:58_5"/>
<span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</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">p</font>, <font color="Olive" title="b3">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="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T49" onmouseover="rs('filter_2/T49')" onmouseout="rh()">FILTER_2:49</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r</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="E4:58_5"/><span class="lab"><font color="Green" title="E40">F3</font></span>: 
( <font color="Maroon" title="c11">r</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">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="c11">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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="c8">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_5"><span class="lab"><font color="Green" title="E39">f2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">q</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="E6:58_5"/><span class="lab"><font color="Green" title="E41">F4</font></span>: 
(  <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="c12">p</font> <a href="lattices.html#K3" title="LATTICES:func.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="c8">P</font> &amp; <font color="Maroon" title="c13">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_5"><span class="lab"><font color="Green" title="E40">F3</font></span></a></span>;<br/>
<a NAME="E7:58_5"/><span class="lab"><font color="Green" title="E42">F5</font></span>: 
 <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="c12">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T1" onmouseover="rs('filter_0/T1')" onmouseout="rh()">FILTER_0:1</a>, <a class="txt" href="#E6:58_5"><span class="lab"><font color="Green" title="E41">F4</font></span></a>, <a class="ref" href="filter_2.html#T28" onmouseover="rs('filter_2/T28')" onmouseout="rh()">FILTER_2:28</a></span>;<br/>
<a NAME="E8:58_5"/>
<font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI: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> : <font color="Maroon" title="c2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">x</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> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_5"><span class="lab"><font color="Green" title="E42">F5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:58_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:58"><span class="lab"><font color="Green" title="E37">hh</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="#E6:58_5"><span class="lab"><font color="Green" title="E41">F4</font></span></a>, <a class="txt" href="#E18:58"><span class="lab"><font color="Green" title="E30">W0</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="c11">QQ</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="E32:58"/><span class="lab"><font color="Green" title="E39">W1</font></span>: 
( <font color="Maroon" title="c11">QQ</font> is  <a href="filter_2.html#V2" title="FILTER_2:attr.2">prime</a>  &amp; <font color="Maroon" title="c10">Pa</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">QQ</font> &amp;  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="c11">QQ</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="latticea.html#T20" target="_self" onmouseover="rs('latticea/T20')" onmouseout="rh()">Cor16</a></span>;<br/>
<a NAME="E33:58"/>
<font color="Maroon" title="c11">QQ</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="#E32:58"><span class="lab"><font color="Green" title="E39">W1</font></span></a></span>;<br/>
<a NAME="E34:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">w1</font></span>: 
<font color="Maroon" title="c11">QQ</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="E35:58"/><span class="lab"><font color="Green" title="E41">W2</font></span>: 
( <font color="Maroon" title="c8">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c11">QQ</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="latticea.html#K8" title="LATTICEA:func.8">Spectrum</a> <font color="Maroon" title="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58"><span class="lab"><font color="Green" title="E30">W0</font></span></a>, <a class="txt" href="#E32:58"><span class="lab"><font color="Green" title="E39">W1</font></span></a>, <a class="txt" href="#E34:58"><span class="lab"><font color="Green" title="E40">w1</font></span></a>, <a class="txt" href="#E20:58"><span class="lab"><font color="Green" title="E31">w0</font></span></a></span>;<br/>
<a NAME="E36:58"/><span class="lab"><font color="Green" title="E42">W3</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:58"><span class="lab"><font color="Green" title="E38">zx</font></span></a>, <a class="txt" href="#E32:58"><span class="lab"><font color="Green" title="E39">W1</font></span></a>, <a class="txt" href="#E25:58"><span class="lab"><font color="Green" title="E35">ZM</font></span></a>, <a class="txt" href="#E22:58"><span class="lab"><font color="Green" title="E32">ZZ</font></span></a>, <a class="txt" href="#E24:58"><span class="lab"><font color="Green" title="E34">ZK</font></span></a></span>;<br/>
<a NAME="E37:58"/>
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">QQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:58"><span class="lab"><font color="Green" title="E39">W1</font></span></a>, <a class="txt" href="#E23:58"><span class="lab"><font color="Green" title="E33">ZL</font></span></a>, <a class="txt" href="#E25:58"><span class="lab"><font color="Green" title="E35">ZM</font></span></a></span>;<br/>
<a NAME="E38:58"/><span class="kw">then </span>
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c11">QQ</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E39:58"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E23">d1</font></span></a>, <a class="txt" href="#E35:58"><span class="lab"><font color="Green" title="E41">W2</font></span></a>, <a class="txt" href="#E36:58"><span class="lab"><font color="Green" title="E42">W3</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/>


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