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<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="dtconstr.html#K12" title="DTCONSTR:func.12">PN-to-NAT</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K14" title="DTCONSTR:func.14">NAT-to-PN</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>;<br/>
<a NAME="E1:51"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="dtconstr.html#E45"><span class="lab"><font color="Green" title="E34">Lm17</font></span></a></span>;<br/>
<a NAME="E2:51"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="dtconstr.html#E46"><span class="lab"><font color="Green" title="E35">Lm18</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:51"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E37">A2</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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