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

<b>set </b><font color="Maroon">Y</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #);<br/>
<a NAME="E1:107_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:107_1_1_1"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) <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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:107_1_1_1_1"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:107_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) <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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:107_1_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">FF</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)  st <font color="Olive">FF</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) holds <br/> <a href="tarski.html#K3">union</a> <font color="Olive">FF</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">FF</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #);<br/>

<b>set </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">SAA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> </span>}</span> ;<br/>
<a NAME="E1:107_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">SAA</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">FF1</font> = <span class="p1">{<span class="default"> <font color="Olive">H</font> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span> ;<br/>
<a NAME="E3:107_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">H</font> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">H</font> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span> 
 ;<br/><a NAME="E2:107_1_1_1_2_1_1"/><b>then </b>
 ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">H</font> &amp; <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) )
 ;<br/><b>hence </b><a NAME="E3:107_1_1_1_2_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">H</font> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF1</font> = <span class="p1">{<span class="default"> <font color="Olive">H</font> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">FF2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span> ;<br/>
<a NAME="E5:107_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span> 
 ;<br/><a NAME="E2:107_1_1_1_2_2_1"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) )
 ;<br/><b>hence </b><a NAME="E3:107_1_1_1_2_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp;  ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<b>assume </b><a NAME="E7:107_1_1_1_2"/>
<font color="Maroon">FF</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 ;<br/>

<a NAME="E8:107_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">FF</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 
;<br/>
<a NAME="E9:107_1_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T15" target="_self">Th15</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E10:107_1_1_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_1_1_2_3_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#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="107_1_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_3_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:107_1_1_1_2_3_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">SAA</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:107_1_1_1_2_3_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">SAA</font> <a href="tarski.html#R1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 ;<br/><a NAME="E2:107_1_1_1_2_3_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:107_1_1_1_2_3_1_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</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>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_1_1_1"/>
<font color="Maroon">SAA</font> <a href="tarski.html#R1">c=</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>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:107_1_1_1_2_3_1_1_1_1"/>
<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> <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_3_1_1_1_1_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E2:107_1_1_1_2_3_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp;  <a href="xboole_0.html#K1">{}</a>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:107_1_1_1_2_3_1_1_1_1_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_1_1_2"/>
<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> <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:107_1_1_1_2_3_1_1_1_1"/>
<font color="Maroon">SAA</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>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_1_1_2_3_1_1_1_2_1"/>
( <font color="Maroon">FF2</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">FF2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_3_1_1_1_2_1_1"/>
<font color="Maroon">FF2</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_1_2_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_3_1_1_1_2_1_2"/>
<font color="Maroon">FF2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_1_2_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:107_1_1_1_2_3_1_1_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_3_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:107_1_1_1_2_3_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10">TSEP_1:10</a></i>;<br/><a NAME="E4:107_1_1_1_2_3_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E5:107_1_1_1_2_3_1_1_2"/><i><font color="Green">E48</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SAA</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:107_1_1_1_2_3_1_1_2_1"/><i><font color="Green">E49</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_3_1_1_2_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E3:107_1_1_1_2_3_1_1_2_1_1_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>hence </b><a NAME="E4:107_1_1_1_2_3_1_1_2_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_2_1_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">SAA</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:107_1_1_1_2_3_1_1_2_1"/>
<font color="Maroon">SAA</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_3_1_1_2_1_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 ;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:107_1_1_1_2_3_1_1_2_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:107_1_1_1_2_3_1_1_2_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E6:107_1_1_1_2_3_1_1_2_1_2_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>hence </b><a NAME="E7:107_1_1_1_2_3_1_1_2_1_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_3_1_1_2_1_2"/>
<font color="Maroon">SAA</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:107_1_1_1_2_3_1_1_2_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SAA</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">SS</font> = <font color="Maroon">FF2</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/><a NAME="E7:107_1_1_1_2_3_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">SS</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><b>hence </b><a NAME="E8:107_1_1_1_2_3_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 <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="E2:107_1_1_1_2_3"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SAA</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:107_1_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:107_1_1_1_2_4"/><i><font color="Green">E53</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 ;<br/><b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:107_1_1_1_2_4_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:107_1_1_1_2_4_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:107_1_1_1_2_4_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E7:107_1_1_1_2_4_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 ;<br/><a NAME="E8:107_1_1_1_2_4_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:107_1_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:107_1_1_1_2_4_1_1"/>
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:107_1_1_1_2_4_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E12:107_1_1_1_2_4_1_1"/>
(  <a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> &amp;  <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E13:107_1_1_1_2_4_1_1"/><b>then </b>
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:107_1_1_1_2_4_1_1"/><b>then </b>
( <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E15:107_1_1_1_2_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_4_1"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:107_1_1_1_2_4"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1_2_4_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:107_1_1_1_2_4_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_4_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:107_1_1_1_2_4_2_1_1_1_1"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:107_1_1_1_2_4_2_1_1_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E6:107_1_1_1_2_4_2_1_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:107_1_1_1_2_4_2_1_1_1_1"/><i><font color="Green">E178</font></i>: 
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:107_1_1_1_2_4_2_1_1_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/><a NAME="E10:107_1_1_1_2_4_2_1_1_1_1"/>
( <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E11:107_1_1_1_2_4_2_1_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E12:107_1_1_1_2_4_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:107_1_1_1_2_4_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:107_1_1_1_2_4_2_1_1_1_2"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:107_1_1_1_2_4_2_1_1_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:107_1_1_1_2_4_2_1_1_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:107_1_1_1_2_4_2_1_1_1_2"/><i><font color="Green">E182</font></i>: 
 ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:107_1_1_1_2_4_2_1_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/><a NAME="E10:107_1_1_1_2_4_2_1_1_1_2"/>
( <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E11:107_1_1_1_2_4_2_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E12:107_1_1_1_2_4_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:107_1_1_1_2_4_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:107_1_1_1_2_4_2"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:107_1_1_1_2_4"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">FF2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:107_1_1_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">FF</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="tmap_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, <a class="ref" href="tmap_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:107_1_1_1"/>
for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)  st <font color="Olive">C</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) &amp; <font color="Olive">D</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) holds <br/><font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">D</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #), <font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:107_1_1_1_3"/>
( <font color="Maroon">C</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) &amp; <font color="Maroon">D</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) )
 ;<br/>

<a NAME="E2:107_1_1_1_3"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span>  &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span>  )
 
;<br/>
<b>then consider </b><font color="Maroon">H1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">G1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:107_1_1_1_3"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E5:107_1_1_1_3"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 ;<br/>
<b>consider </b><font color="Maroon">H2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">G2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:107_1_1_1_3"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E8:107_1_1_1_3"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:107_1_1_1_3"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E10:107_1_1_1_3"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E11:107_1_1_1_3"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E12:107_1_1_1_3"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E13:107_1_1_1_3"/>
( <font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:107_1_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E15:107_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E16:107_1_1_1_3"/><b>then </b><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">H2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/>
<a NAME="E18:107_1_1_1_3"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E19:107_1_1_1_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</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>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #)
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:107_1_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) is   <a href="pre_topc.html#NM1">TopSpace</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T14" target="_self">Th14</a>, <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="E2:107_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font></span>)</span> #) is  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>
 ;<br/>


</div>
