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<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> <a href="algseq_1.html#K2" title="ALGSEQ_1:func.2">PSeg</a> $1 is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>;<br/>
<a NAME="E1:2"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="algseq_1.html#T11" title="ALGSEQ_1:th.11">ALGSEQ_1:11</a></i>;<br/>
<a NAME="E2:2"/><i><font color="Green" title="E4">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">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>assume </b><a NAME="E1:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 ;<br/>

<a NAME="E2:2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K2" title="ALGSEQ_1:func.2">PSeg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>
<b>hence </b><a NAME="E3:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="algseq_1.html#T17" title="ALGSEQ_1:th.17">ALGSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:2"/>
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="dynkin.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="dynkin.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>);</i><br/>


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