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<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A1</font> is <a href="prob_1.html#V3">non-decreasing</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><span class="p1">(<span class="default"><a href="prob_3.html#K2">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:23"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="prob_3.html#D2">PROB_3:def 2</a></i>;<br/>
<a NAME="E3:23"/><i><font color="Green">E39</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="23_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="E1:23_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:23_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <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="prob_2.html#T15">PROB_2:15</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="prob_3.html#K2">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#D2">PROB_3:def 2</a>, </i>
<br/>.= 
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <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="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:23"/>
for <font color="Olive">k</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">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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