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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a></span><br/>

<b>set </b><font color="Maroon" title="c2">B</font> =  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18" target="_self" title="LATTICE3:def.18">LATTICE3:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span> st <br/>( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">p</font> &amp; ( for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b2">r</font> holds <br/><font color="Olive" title="b1">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b2">r</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c4">p</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:56"/>
<font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a>, <a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></i>;<br/>
<a NAME="E3:56"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self" title="LATTICE3:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="lattice3.html#D1" target="_self" title="LATTICE3:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">p</font> &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">r</font> ) )</span><br/>

<b>thus </b><a NAME="E5:56"/>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">p</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">r</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c6">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font> )</span><br/>

<b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font></span><br/>

<a NAME="E2:56_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:56_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E4:56_1"/>
<font color="Maroon" title="c6">q</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self" title="LATTICE3:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c6">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">r</font> implies <font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">r</font> )</span><br/>

<b>assume </b><a NAME="E6:56"/><i><font color="Green" title="E34">A3</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> holds <br/><font color="Olive" title="b1">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">r</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span> implies <font color="Maroon" title="c7">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">r</font> )</span><br/><b>assume </b><a NAME="E1:56_2"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">r</font></span><br/><a NAME="E2:56_2"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">z'</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="#E1:56_2"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:56_2"/>
<font color="Maroon" title="c8">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="txt" href="#E6:56"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E2:56_2"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E5:56_2"/>
<font color="Maroon" title="c7">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self" title="LATTICE3:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:56"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94" title="ZFMISC_1:th.94">ZFMISC_1:94</a></i>;<br/>
<b>hence </b><a NAME="E9:56"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self" title="LATTICE3:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
