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<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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <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="E1:27"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">A</font> is    <a href="connsp_3.html#M1" target="_self">a_union_of_components</a> of <font color="Maroon">GX</font> &amp; <font color="Maroon">B</font> is    <a href="connsp_3.html#M1" target="_self">a_union_of_components</a> of <font color="Maroon">GX</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">Fa</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E13</font></i>: 
( ( for <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fa</font> holds <br/><font color="Olive">B2</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fa</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Fb</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green">E25</font></i>: 
( ( for <font color="Olive">B3</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fb</font> holds <br/><font color="Olive">B3</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> ) &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fb</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Fd</font> = <font color="Maroon">Fa</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Fb</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font> ;<br/>
<a NAME="E7:27"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:27_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<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="E4:27_1_1"/><i><font color="Green">E30</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">Fa</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i><a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fb</font>
 ;<br/><a NAME="E2:27_1_1_1"/><b>then </b>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:27_1_1"/><b>then </b>
<font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fd</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_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="E1:27_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</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="E3:27_1_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">Fd</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:27_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fa</font> &amp; not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fb</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i/>;<br/>
<a NAME="E6:27_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">A2</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 
<b>by </b><i>, <a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E7:27_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">Y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y3</font> &amp; <font color="Maroon">Y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fb</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">B3</font> = <font color="Maroon">Y3</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> <b>by </b><i><a class="txt" href="connsp_3.html#E1:27_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E5:27_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">B3</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:27"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_3.html#E1:27_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E6:27_1_2_1"/>
<font color="Maroon">A2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="connsp_3.html#E1:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:27_1_2_1"/><b>then </b>
<font color="Maroon">A2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B3</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E8:27_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_3.html#E5:27"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_3.html#E7:27"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_3.html#E1:27_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="connsp_1.html#T37">CONNSP_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:27_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Fd</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>
<a NAME="E8:27"/>
for <font color="Olive">B4</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fd</font> holds <br/><font color="Olive">B4</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="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B4</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>

<b>assume </b><a NAME="E1:27_2"/>
<font color="Maroon">B4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fd</font>
 ;<br/>

<a NAME="E2:27_2"/><b>then </b>
<font color="Maroon">B4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fa</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon">B4</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:27"/>
<font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</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>
