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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">filled</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K2" title="FINTOPO3:func.2">Fcl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E2:24"/><i><font color="Green" title="E20">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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" title="E21">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="fintopo3.html#D2" target="_self" title="FINTOPO3:def.2">Def2</a></i>;<br/>
<a NAME="E2:24_1"/><i><font color="Green" title="E22">A4</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/>

<a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K3" title="FINTOPO3:func.3">Fcl</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">k</font></span>)</span> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T19" title="FIN_TOPO:th.19">FIN_TOPO:19</a></i>;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K3" title="FINTOPO3:func.3">Fcl</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T13" target="_self" title="FINTOPO3:th.13">Th13</a></i>;<br/>
<a NAME="E4:24_1_1"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K2" title="FINTOPO3:func.2">Fcl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo3.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="fin_topo.html#T18" title="FIN_TOPO:th.18">FIN_TOPO:18</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="fintopo3.html#E4:24_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="fintopo3.html#E1:24_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="fintopo3.html#E2:24_1"><i><font color="Green" title="E22">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:24_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>thus </b><a NAME="E3:24"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K3" title="FINTOPO3:func.3">Fcl</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="fintopo3.html#E2:24"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>


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