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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:26"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="prob_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="prob_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> 0
<b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E2:26_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:26_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p3">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</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">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E1:26_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="prob_3.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</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">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:26_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prob_3.html#E2:26_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="prob_3.html#E3:26_1"><i><font color="Green">E28</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26"/>
 <a href="prob_3.html#K2" target="_self">Partial_Union</a> <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
