<?xml version="1.0"?>
<div class="add">

<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#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green">E23</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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green">E24</font></i>: 
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> <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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a>  ) )
 ;<br/>

<b>consider </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><br/><a NAME="E4:28"/><i><font color="Green">E25</font></i>: 
( <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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green">E26</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>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> not <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</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> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="E1:28_1"/><i><font color="Green">E27</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:28_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/><a NAME="E3:28_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:28_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:28_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E4:28"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E6:28_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:28"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</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="E1:28_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:28_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>assume </b><a NAME="E3:28_1_1_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E4:28_1_1_1"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E5:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="connsp_1.html#E2:28_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E5:28_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:28_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:28_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E5:28_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="connsp_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E8:28_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="connsp_1.html#E3:28_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="connsp_1.html#E4:28_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_1_1"/><b>then </b>
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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:28_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><a NAME="E5:28_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E6:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E4:28_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E7:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E8:28_1_1"/>
<font color="Maroon">c<sub>6</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#E4:28_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</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="E1:28_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:28_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:28_1_2_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E4:28_1_2_1"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E5:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="connsp_1.html#E2:28_1_2_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E5:28_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:28_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:28_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E5:28_1_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="connsp_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E8:28_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E1:28_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="connsp_1.html#E3:28_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="connsp_1.html#E4:28_1_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><a NAME="E3:28_1_2"/><b>then </b>
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>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:28_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><a NAME="E5:28_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E6:28_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:28_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E4:28_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E7:28_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E8:28_1_2"/>
<font color="Maroon">c<sub>5</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#E4:28_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:28_1"/>
( <font color="Maroon">c<sub>5</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>6</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#E2:28_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E5:28_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="connsp_1.html#E6:28_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="connsp_1.html#E7:28_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="connsp_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:28"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T16" target="_self">Th16</a></i>;<br/>


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