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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="fin_topo.html#L1">FT_Space_Str</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/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="fin_topo.html#V2">filled</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K2" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:24"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="fintopo3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E2:24_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K3" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K3" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/>
<a NAME="E4:24_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K3" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:24_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K2" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E6:24_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo3.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fin_topo.html#T18">FIN_TOPO:18</a></i>;<br/>
<b>hence </b><a NAME="E7:24_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="fintopo3.html#E5:24_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fintopo3.html#E1:24_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fintopo3.html#E2:24_1"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:24_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E3:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K2" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo3.html#E2:24"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K3" target="_self">Fcl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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