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

<b>let </b><font color="Maroon">GX</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Fu</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> holds <br/><font color="Olive">A</font> is    <a href="connsp_3.html#M1" target="_self">a_union_of_components</a> of <font color="Maroon">GX</font>
 ;<br/>

<a NAME="E2:25"/>
<span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> :  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> :  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp;  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">GX</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F2</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> :  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> ;<br/>
<a NAME="E4:25"/><i><font color="Green">E13</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<a NAME="E1:25_2"/><i><font color="Green">E25</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_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="E1:25_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B3</font> = <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/>
<a NAME="E5:25_2_1"/>
<font color="Maroon">B3</font> is    <a href="connsp_3.html#M1" target="_self">a_union_of_components</a> of <font color="Maroon">GX</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E7:25_2_1"/><i><font color="Green">E28</font></i>: 
( ( for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) &amp; <font color="Maroon">B3</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:25_2_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A3</font> = <font color="Maroon">Y2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/>
<a NAME="E11:25_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A3</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:25_2_1"/>
<font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B3</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E13:25_2_1"/><b>then </b>
<font color="Maroon">A3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="txt" href="connsp_3.html#E1:25_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:25_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2_2"><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="E1:25_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_2_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:25_2_2"/>
 ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp;  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:25_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:25"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25"/>
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> holds <br/><font color="Olive">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>

<b>assume </b><a NAME="E1:25_3"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font>
 ;<br/>

<a NAME="E2:25_3"/><b>then </b>
 ex <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> &amp;  ex <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fu</font> &amp; <font color="Olive">A2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B2</font> &amp; <font color="Olive">A2</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:25_3"/>
<font color="Maroon">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fu</font> is    <a href="connsp_3.html#M1" target="_self">a_union_of_components</a> of <font color="Maroon">GX</font>
 <b>by </b><i>, </i>;<br/>


</div>
