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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">T</font> is <a href="fin_topo.html#V1">filled</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K6" target="_self">Finf</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:39"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E2:39_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K7" target="_self">Finf</a> <font color="Maroon">A</font>,<font color="Maroon">k</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:39_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a>  <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K7" target="_self">Finf</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:39_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K6" target="_self">Finf</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:39_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i><a class="ref" href="fintopo3.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E6:39_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="fintopo3.html#E2:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:39_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fintopo3.html#E1:39_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="fintopo3.html#T4" target="_self">Th4</a>);<br/></i>
<b>hence </b><a NAME="E4:39_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E3:39"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K7" target="_self">Finf</a> <font color="Maroon">A</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="fintopo3.html#T3" target="_self">Th3</a></i>;<br/>


</div>
