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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:6"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green">E7</font></i>: 
 <a href="connsp_3.html#K1" target="_self">skl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="connsp_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:6"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:6"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E3:6"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E7:6"/>
<font color="Maroon">c<sub>3</sub></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#E1:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="connsp_3.html#E3:6"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E8:6"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E5:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<a NAME="E9:6"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:6"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:6"/>
 <a href="connsp_3.html#K1" target="_self">skl</a> <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="connsp_3.html#E4:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="connsp_3.html#E6:6"><i><font color="Green">E9</font></i></a>, <a class="ref" href="connsp_1.html#T27">CONNSP_1:27</a></i>;<br/>


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