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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>;<br/>
<a NAME="E1:30_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">XSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self" title="PROB_3:def.1">Def1</a></i>;<br/>
<a NAME="E2:30_1_1"/><b>then </b><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<a NAME="E3:30_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 ;<br/>

<a NAME="E2:30_1_1_1"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">k</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E3:30_1_1_1"/>
<font color="Maroon" title="c3">XSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<a NAME="E4:30_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">XSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E5:30_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">XSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="txt" href="prob_3.html#E2:30_1_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="prob_1.html#T49" title="PROB_1:th.49">PROB_1:49</a></i>;<br/>
<a NAME="E6:30_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">XSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Si</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self" title="PROB_3:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:30_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="prob_3.html#E2:30_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="prob_3.html#E3:30_1_1"><i><font color="Green" title="E31">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:30_1_1"/>
 <a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c3">XSeq</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>


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