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



<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:9_1"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>2</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> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E3:9_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <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>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">skl</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="connsp_3.html#E2:9_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="connsp_3.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:9_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E2:9_2"/><i><font color="Green">E10</font></i>: 
<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>and </b><br/><a NAME="E3:9_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">skl</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:9_2"/><i><font color="Green">E12</font></i>: 
<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#E1:9_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="connsp_3.html#E2:9_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="connsp_3.html#E3:9_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="connsp_3.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:9_2"/>
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> 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> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:9_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="connsp_3.html#E3:9_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="connsp_3.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E6:9_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="connsp_3.html#E4:9_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> iff ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">skl</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:9"><i><font color="Green">E8</font></i></a></i>;<br/>


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