<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> implies  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) );<br/>
<a NAME="E1:18"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> 0
 ;<br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E2:18_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> 0 )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> implies  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:18_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:18_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1_1"><b>case </b></a><a NAME="E1:18_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:18_2_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><a NAME="E4:18_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E5:18_2_1_1_1"/>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1_2"><b>case </b></a><a NAME="E1:18_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_1_1_2"/>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_2"/>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:18"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> implies  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) )
 ;<br/>

<b>given </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E5:18"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E7:18"/>
<font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:18"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E10:18"/>
 <a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font> is <a href="prob_1.html#V3">non-decreasing</a>
 
<b>by </b><i/>;<br/>
<a NAME="E11:18"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#D7">PROB_1:def 7</a>, </i>;<br/>
<b>hence </b><a NAME="E12:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div>
