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

<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">l</font> is <a href="waybel_6.html#V5" target="_self">prime</a> implies for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>  st ( for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> ) ) holds <br/>( <font color="Olive">f</font> is <a href="waybel_0.html#V19">meet-preserving</a> &amp; <font color="Olive">f</font> is <a href="waybel_0.html#V20">join-preserving</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">l</font> is <a href="waybel_6.html#V5" target="_self">prime</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>



<b>assume </b><a NAME="E2:44_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 ;<br/>

<a NAME="E3:44_1"/><i><font color="Green">E48</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E50</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>
;<br/>
<a NAME="E5:44_1"/>
for <font color="Olive">z</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">z</font>,<font color="Olive">y</font></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:44_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E2:44_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font> implies  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_2_1_1"/>
( ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:44_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:44_1_2_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:44_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> &amp; not <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:44_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:44_1_2_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_2_1_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:44_1_2_1_1_3"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:44_1_2_1_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:44_1_2_1_1_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:44_1_2_1_1_4"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E3:44_1_2_1_1_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
;<br/>
<b>hence </b><a NAME="E5:44_1_2"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:44_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V19">meet-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a></i>;<br/>

<a NAME="E7:44_1"/>
for <font color="Olive">z</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">z</font>,<font color="Olive">y</font></span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:44_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E2:44_1_3"/><b>then </b><i><font color="Green">E91</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font> implies  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a></i>;<br/>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_3_1_1"/>
( ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_3_1_1_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:44_1_3_1_1_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/><a NAME="E3:44_1_3_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_3_1_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:44_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; not <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E3:44_1_3_1_1_2"/><b>then </b>
not <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E4:44_1_3_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_3_1_1_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:44_1_3_1_1_3"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E3:44_1_3_1_1_3"/><b>then </b>
not <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E4:44_1_3_1_1_3"/><b>then </b><i><font color="Green">E108</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:44_1_3_1_1_4"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:44_1_3_1_1_4"/><b>then </b>
( <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; not <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E3:44_1_3_1_1_4"/><b>then </b>
not <font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E4:44_1_3_1_1_4"/><b>then </b><i><font color="Green">E111</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
;<br/>
<b>hence </b><a NAME="E5:44_1_3"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:44_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V20">join-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a></i>;<br/>


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

<b>thus </b><a NAME="E2:44"/>
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>  st ( for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> ) ) holds <br/>( <font color="Olive">f</font> is <a href="waybel_0.html#V19">meet-preserving</a> &amp; <font color="Olive">f</font> is <a href="waybel_0.html#V20">join-preserving</a> ) ) implies <font color="Maroon">l</font> is <a href="waybel_6.html#V5" target="_self">prime</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>  st ( for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> ) ) holds <br/>( <font color="Olive">f</font> is <a href="waybel_0.html#V19">meet-preserving</a> &amp; <font color="Olive">f</font> is <a href="waybel_0.html#V20">join-preserving</a> )
 ;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6" target="_self">WAYBEL_6:def 6</a><br/>

<b>assume </b><a NAME="E2:44_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> ;<br/>
<i><font color="Green">E114</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> iff <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E5:44_2"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2_1_1"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> or  not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><b>reconsider </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:44_2_2_1_1_1"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> iff <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:44_2_2_1_1_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_2"/><i><font color="Green">E119</font></i>: 
not <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">b</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/><b>reconsider </b><font color="Maroon">b</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:44_2_2_1_1_2"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> iff <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:44_2_2_1_1_2"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:44_2_2"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span><b> such that </b><br/><a NAME="E7:44_2"/><i><font color="Green">E120</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10">FUNCT_2:sch 10</a>();<br/></i>
<a NAME="E8:44_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:44_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<a NAME="E10:44_2"/><i><font color="Green">E122</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/>
<a NAME="E11:44_2"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V19">meet-preserving</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:44_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a></i>;<br/>
<i><font color="Green">E124</font></i>:  <a href="xboole_0.html#K1">{}</a>  = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i>, , , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_4"/>
( not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E2:44_2_4"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:44_2_4"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:44_2"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> or <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/>


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


</div>
