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

<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>, <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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</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></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:43"/><i><font color="Green">E29</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="43_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="fintopo3.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="fintopo3.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="fintopo3.html#D8" target="_self">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E2:43_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:43_1"/><i><font color="Green">E31</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="43_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:43_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="fintopo3.html#K9" target="_self">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> 
<b>by </b><i><a class="ref" href="fintopo3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> 
<b>by </b><i><a class="txt" href="fintopo3.html#E1:43_1_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> </span>)</span>
<b>by </b><i><a class="ref" href="fintopo3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="fintopo3.html#K1" target="_self">^d</a> </span>)</span>
<b>by </b><i><a class="ref" href="fintopo3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fintopo3.html#T38" target="_self">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:43_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:43_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#E1:43_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="fintopo3.html#E2:43_1"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:43_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>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E2:43"/>
<span class="p1">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo3.html#E1:43"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>then </b> <a href="fintopo3.html#K9" target="_self">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:43"/>
 <a href="fintopo3.html#K9" target="_self">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" target="_self">Fdfl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


</div>
