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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>given </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:33"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Maroon">a</font>,<font color="Olive">x</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 ;<br/>

<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/><a NAME="E1:33_1"/>
<font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 <b>by </b><i><a class="ref" href="connsp_1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E2:33_1"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">C1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">C1</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/><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="E2:33_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i><b>take </b><font color="Maroon">F</font> = <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/><b>thus </b><a NAME="E3:33_2"/>
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies ( <font color="Maroon">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:33_2_1"/><b>then </b>
 ex <font color="Olive">C1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Olive">C1</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:33_2_1"/>
( <font color="Maroon">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:33_2"/>
( <font color="Maroon">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<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="E5:33"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">C</font> ) )
 ;<br/>
<a NAME="E6:33"/><i><font color="Green">E29</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">F</font> holds <br/><font color="Olive">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><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="E1:33_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font>
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:33_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E4:33_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:33_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E6:33"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E6:33_3"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:33_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="connsp_1.html#E5:33_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="connsp_1.html#E9:33"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E7:33_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:33"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:33"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:33"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:33"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E13:33"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E9:33"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E14:33"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 
<b>by </b><i>, , <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E15:33"/>
<font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:33"><i><font color="Green">E24</font></i></a></i>;<br/>


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