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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V1" target="_self">lower</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">BB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>set </b><font color="Maroon">s</font> =  <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#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>;<br/><a NAME="E4:14_1"/><i><font color="Green">E36</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">S</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> ,<font color="Maroon">T</font> &amp;  <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">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1"/>
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font> or <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">TOPS_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:14_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:14_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/><a NAME="E4:14_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/><a NAME="E5:14_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/><a NAME="E6:14_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_1_1_2_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:14_1_1_2_1_1"/>
 <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#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:14_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E4:14_1_1_2_1"/><i><font color="Green">E65</font></i>: 
 <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#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E5:14_1_1_2_1"/>
<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"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">a</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:14_1_1_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T8">YELLOW_3:8</a></i>;<br/>
<a NAME="E7:14_1_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>
<a NAME="E8:14_1_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E9:14_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E10:14_1_1_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>hence </b><a NAME="E11:14_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:14_1_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T35">YELLOW_9:35</a></i>;<br/>


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