<?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="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E23</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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</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>5</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<i><font color="Green">E24</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <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> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a></i>
;<br/>
<a NAME="E7:26"/><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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/><a NAME="E2:26_2"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font> 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/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <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> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:26_2"/><i><font color="Green">E27</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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:26_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:26_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:26_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E8:26_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E9:26_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E11:26_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E12:26_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E7:26_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</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="E14:26_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E15:26_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E8:26_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E6:26"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E18:26_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E11:26_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E19:26_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E20:26_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E12:26_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E21:26_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E22:26_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:26_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E12:26_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="connsp_1.html#E21:26_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E24:26_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E25:26_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E20:26_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E26:26_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E18:26_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E28:26_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E14:26_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E29:26_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E30:26_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E15:26_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E31:26_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E32:26_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:26_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E15:26_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="connsp_1.html#E31:26_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E34:26_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E35:26_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E30:26_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E36:26_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E28:26_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E37:26_2"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>8</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E7:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E4:26_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E5:26_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E6:26_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E9:26_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="connsp_1.html#E26:26_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="connsp_1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E38:26_2"/>
not <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="ref" href="connsp_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/><a NAME="E2:26_3"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font> 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/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:26_3"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:26_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:26_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:26_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E8:26_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E9:26_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T11" target="_self">Th11</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:26_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E12:26_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E7:26_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:26_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E15:26_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E8:26_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</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="E17:26_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E18:26_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E11:26_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</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="E20:26_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E21:26_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E14:26_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>9</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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E6:26"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E6:26"><i><font color="Green">E24</font></i></a></i>;<br/><font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E12:26_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="connsp_1.html#E18:26_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/><a NAME="E25:26_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E17:26_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E15:26_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="connsp_1.html#E21:26_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/><a NAME="E27:26_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E20:26_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E28:26_3"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>7</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E7:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E4:26_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E5:26_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="connsp_1.html#E6:26_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E9:26_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="connsp_1.html#E25:26_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="connsp_1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E29:26_3"/>
not <font color="Maroon">c<sub>3</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="E10:26"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a> iff <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#E8:26"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
