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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fin_topo.html#L1">FT_Space_Str</a> ;<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>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="fin_topo.html#V6">connected</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="fintopo4.html#R2" target="_self">is_continuous</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> &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="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D17">FIN_TOPO:def 17</a></i>;<br/>
<a NAME="E3:18"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> &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="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E1:18_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</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>7</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>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E4:18_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fintopo4.html#E3:18_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E3:18_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <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/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K9">/\</a> <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/>
<a NAME="E9:18_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E10:18_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:18_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18"><i><font color="Green">E9</font></i></a>, <a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="relat_1.html#T175">RELAT_1:175</a></i>;<br/>
<a NAME="E12:18_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E10:18_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E13:18_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E14:18_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E15:18_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>10</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="E16:18_1"/>
<font color="Maroon">c<sub>12</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="E17:18_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E15:18_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E18:18_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E14:18_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E19:18_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E20:18_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>12</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="E21:18_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E19:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E23:18_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E24:18_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:18_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="fintopo4.html#E1:18"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E27:18_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E23:18_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fintopo4.html#E26:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E28:18_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</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="fintopo4.html#E26:18_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E30:18_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E31:18_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E33:18_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="fintopo4.html#E1:18"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E34:18_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E30:18_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fintopo4.html#E33:18_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E35:18_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</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="fintopo4.html#E33:18_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E36:18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E37:18_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T171">RELAT_1:171</a></i>;<br/>
<a NAME="E38:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>;<br/>
<a NAME="E39:18_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E21:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E40:18_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E41:18_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fintopo4.html#E18:18_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fintopo4.html#E28:18_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fintopo4.html#E35:18_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E42:18_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</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="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E43:18_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E19:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/>
<a NAME="E44:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E20:18_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E45:18_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</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="fintopo4.html#E42:18_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E46:18_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E47:18_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <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="fintopo4.html#E45:18_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<a NAME="E48:18_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T154">RELAT_1:154</a></i>;<br/>
<a NAME="E49:18_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <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="fintopo4.html#E47:18_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E50:18_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>
<a NAME="E51:18_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E52:18_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:18"><i><font color="Green">E9</font></i></a>, <a class="ref" href="fintopo4.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E53:18_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E5:18_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E54:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E55:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E51:18_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E56:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</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="fintopo4.html#E49:18_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E57:18_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="fin_topo.html#V6">connected</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D17">FIN_TOPO:def 17</a></i>;<br/>


</div>
