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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#NM1">Element</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="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green">E7</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  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:9"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:9"/><i><font color="Green">E9</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="9_1"><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="E1:9_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<a NAME="E2:9_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo5.html#T7" target="_self">Th7</a>, <a class="txt" href="fintopo5.html#E1:9"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:9"/>
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="fintopo5.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="fintopo5.html#E3:9"><i><font color="Green">E9</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:9"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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