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<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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:31_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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="E2:31_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E3:31_1_1"/><i><font color="Green">E31</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="31_1_1_1"><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="E1:31_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:31_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E3:31_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <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> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E4:31_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <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> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E5:31_1_1_1"/><b>then </b>
<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>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:31_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="prob_1.html#T51">PROB_1:51</a></i>;<br/>
<a NAME="E6:31_1_1_1"/><b>then </b>
<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>3</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31_1_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#E2:31_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="prob_3.html#E3:31_1_1"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:31_1_1"/>
 <a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>


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