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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">O</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">o</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Olive">o</font> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> =  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">O</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #);<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E3:56"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">O</font>
 ;<br/>

<a NAME="E4:56"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) is <a href="pre_topc.html#V2">TopSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:56_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:56_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">a</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E4:56_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i>, <a class="ref" href="topgen_2.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:56_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #);<br/><b>assume </b><a NAME="E1:56_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">b</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:56_1_2_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="mssubfam.html#T2">MSSUBFAM:2</a></i>;<br/><a NAME="E4:56_1_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">o</font> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) is   <a href="pre_topc.html#NM1">TopSpace</a> )
 <b>by </b><i><a class="txt" href="topgen_2.html#E1:56_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topgen_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E5:56_1_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:56_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #);<br/>

<b>assume </b><a NAME="E4:56_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_1_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">c</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:56_1_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">o</font> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) is   <a href="pre_topc.html#NM1">TopSpace</a> )
 <b>by </b><i>, <a class="ref" href="topgen_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E4:56_1_3"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:56_1"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:56"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">TT</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>thus </b><a NAME="E7:56"/>
for <font color="Olive">o</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Olive">o</font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">S</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) as   <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, <a class="ref" href="topgen_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:56_2"/>
<font color="Maroon">S</font> is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a><br/>

<b>thus </b><a NAME="E2:56_2_1"/>
the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="topgen_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mssubfam.html#T2">MSSUBFAM:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_2"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #)
 ;<br/>


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

<b>let </b><font color="Maroon">T'</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E8:56"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:56"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">o</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/> <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Olive">o</font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of <font color="Maroon">T'</font>
 ;<br/>

<a NAME="E10:56"/>
<font color="Maroon">TT</font> is    <a href="yellow_9.html#M2">TopExtension</a> of <font color="Maroon">T'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:56_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">TT</font>
 <b>by </b><i><a class="ref" href="topgen_2.html#T2" target="_self">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:56_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">a</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:56_3_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">o</font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of <font color="Maroon">T'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:56_3_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#D5">YELLOW_9:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_3"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T'</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">TT</font>
 <b>by </b><i><a class="ref" href="topgen_2.html#T2" target="_self">Th2</a>, <a class="ref" href="mssubfam.html#T4">MSSUBFAM:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:56"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">B</font> #) is    <a href="yellow_9.html#M2">TopExtension</a> of <font color="Maroon">T'</font>
 ;<br/>


</div>
