<?xml version="1.0"?>
<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="E1:25"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A1</font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/>

<b>thus </b><a NAME="E2:25"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3">Partial_Diff_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="prob_3.html#D3">PROB_3:def 3</a></i>;<br/>

<b>thus </b><a NAME="E3:25"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_3.html#K3">Partial_Diff_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</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="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="prob_3.html#K3">Partial_Diff_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">n</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<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">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#D3">PROB_3:def 3</a></i>
<br/>.= 
<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>


</div>
