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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/>

<a NAME="E1:30"/><i><font color="Green">E25</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E2:30"/><i><font color="Green">E26</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/><a NAME="E2:30_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3" target="_self">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:30_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:30_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E2:30"><i><font color="Green">E26</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E2:30"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E5:30_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_1.html#E1:30_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E2:30_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="connsp_1.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E6:30_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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_1.html#E1:30"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_1.html#E2:30_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E1:30_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:30_2_1"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:30_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/><a NAME="E5:30_2_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:30_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="connsp_1.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E6:30_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:30"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_1.html#E1:30_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E1:30_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><a NAME="E3:30_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E4:30_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a> iff <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:30"><i><font color="Green">E27</font></i></a></i>;<br/>


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