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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X</font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</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> ] <b>means </b><span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font>;<br/>
<a NAME="E2:53"/>
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  <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="setlim_1.html#T23" target="_self" title="SETLIM_1:th.23">Th23</a></i>;<br/>
<a NAME="E2:53_1"/><i><font color="Green" title="E42">A3</font></i>: 
for <font color="Olive" title="b1">k</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>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#K2" title="NAT_1:func.2">+</a> 1]
 
<b>by </b><i><a class="txt" href="setlim_1.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="setlim_1.html#T47" target="_self" title="SETLIM_1:th.47">Th47</a></i>;<br/>
<b>thus </b><a NAME="E3:53_1"/>
for <font color="Olive" title="b1">k</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>1</sub></font>[<font color="Olive" title="b1">k</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="setlim_1.html#E1:53_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="setlim_1.html#E2:53_1"><i><font color="Green" title="E42">A3</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">A1</font>
 ;<br/>


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