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

<b>set </b><font color="Maroon" title="c2">D</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font>;<br/>
<a NAME="E1:36_1_1"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">a</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> , <font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c3">a</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E2:36_1_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  &amp; <font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">H</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:36_1_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c6">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">H</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:36_1_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> &amp; <font color="Maroon" title="c7">H</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 <b>by </b><i><a class="txt" href="tdlat_1.html#E2:36_1_1_1"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/>
<a NAME="E7:36_1_1_1"/>
<font color="Maroon" title="c6">E</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">H</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:36_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="tdlat_1.html#E6:36_1_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tops_1.html#T108" title="TOPS_1:th.108">TOPS_1:108</a></i>;<br/>
<a NAME="E8:36_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">K</font> where <font color="Olive" title="b1">K</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">K</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="tdlat_1.html#E4:36_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="tdlat_1.html#E6:36_1_1_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c4">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">F</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">b</font>
;<br/>

<b>let </b><font color="Maroon" title="c9">A</font>, <font color="Maroon" title="c10">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E10:36_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>

<a NAME="E11:36_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c5">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E12:36_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:36_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">B</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="tdlat_1.html#E1:36_1_1"><i><font color="Green" title="E29">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">h</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K5" title="TDLAT_1:func.5">Closed_Domains_of</a> <font color="Maroon" title="c1">T</font>;<br/>

<b>thus </b><font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> = 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font>
<b>by </b><i><a class="txt" href="tdlat_1.html#E3:36_1_1"><i><font color="Green" title="E30">A5</font></i></a></i>
;<br/>


</div>
