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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="E1:12"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:12"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green">E28</font></i>: 
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:12"/><i><font color="Green">E29</font></i>: 
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:12"/><i><font color="Green">E30</font></i>: 
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>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="E2:12_1"/><i><font color="Green">E43</font></i>: 
( ( for <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">D</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:12_1"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:12_1"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:12"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>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="E2:12_2"/><i><font color="Green">E45</font></i>: 
( ( for <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">D</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D</font> ) ) ) &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:12_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E2:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="connsp_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:12_2"/><b>then </b>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:12_2"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:12"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="connsp_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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