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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">X</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>;<br/><b>set </b><font color="Maroon">T</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>;<br/><a NAME="E1:3_1_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:3_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:3_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">T</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><b>take </b><font color="Maroon">T</font> = <font color="Maroon">T</font>;<br/><b>set </b><font color="Maroon">Y</font> =  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #);<br/><b>thus </b><a NAME="E3:3_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>thus </b><a NAME="E4:3_1_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_2"><b>proof </b></a><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" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #);<br/>

<b>assume </b><a NAME="E1:3_1_1_1_2"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 ;<br/>

<a NAME="E2:3_1_1_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</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">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/>
<a NAME="E3:3_1_1_1_2"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<b>hence </b><a NAME="E4:3_1_1_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" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</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" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #), <font color="Maroon">b</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #);<br/><b>assume </b><b>that </b><br/><a NAME="E5:3_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 <b>and </b><br/><a NAME="E6:3_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 ;<br/><a NAME="E7:3_1_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E8:3_1_1_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" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">X</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">T</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E21</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) &amp; ( for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) holds <br/><font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) ) )
 ;<br/>
<b>take </b>
 <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #)
;<br/>

<b>thus </b><a NAME="E4:3_1_1"/>
not  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>

<b>thus </b><a NAME="E5:3_1_1"/>
(  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) is <a href="pre_topc.html#V1" target="_self">strict</a> &amp;  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">T</font> #) is <a href="pre_topc.html#V2" target="_self">TopSpace-like</a> )
 <b>by </b><i>, </i>;<br/>


</div>
