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<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">FSeq</font> be    <a href="prob_3.html#M1" target="_self">FinSequence</a> of <font color="Maroon">Sigma</font>;<br/>







<b>consider </b><font color="Maroon">ASeq</font> being    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font><b> such that </b><br/><a NAME="E2:68"/><i><font color="Green">E36</font></i>: 
( ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> holds <br/><font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Olive">k</font> ) &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st not <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> holds <br/><font color="Maroon">ASeq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="prob_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:68"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1">Union</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <font color="Maroon">FSeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E4:68"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <font color="Maroon">FSeq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, <a class="ref" href="prob_3.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:68"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <font color="Maroon">FSeq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, <a class="ref" href="prob_3.html#T4" target="_self">Th4</a></i>;<br/>

<a NAME="E6:68"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">FSeq</font> is <a href="prob_2.html#V1">disjoint_valued</a> implies <font color="Maroon">ASeq</font> is <a href="prob_2.html#V1">disjoint_valued</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">FSeq</font> is <a href="prob_2.html#V1">disjoint_valued</a>
 ;<br/>

<a NAME="E2:68_1"/>
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Olive">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:68_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1"/>
( ( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> ) or  not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> )
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_1"/>
<font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="prob_3.html#E1:68_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="prob_3.html#E1:68_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E3:68_1_1_1_1"/><b>then </b>
<font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FSeq</font> <a href="prob_3.html#K11" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T7" target="_self">Th7</a>, <a class="ref" href="prob_3.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:68_1_1_1_1"/>
<font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T7" target="_self">Th7</a>, <a class="ref" href="prob_3.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_1_2"/>
(  not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> or  not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FSeq</font> )
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_2"/><b>then </b>
( <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E3:68_1_1_1_2"/>
<font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">ASeq</font> <a href="kurato_2.html#K7">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1"/>
<font color="Maroon">ASeq</font> is <a href="prob_2.html#V1">disjoint_valued</a>
 <b>by </b><i><a class="ref" href="prob_2.html#D4">PROB_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E7:68"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">FSeq</font> is <a href="prob_2.html#V1">disjoint_valued</a>
 ;<br/>

<b>thus </b><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_3.html#K8" target="_self">Union</a> <font color="Maroon">FSeq</font></span>)</span> = 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1">Union</a> <font color="Maroon">ASeq</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span>
<b>by </b><i>, <a class="ref" href="prob_3.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_3.html#K13" target="_self">*</a> <font color="Maroon">FSeq</font></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#T5" target="_self">Th5</a>, <a class="ref" href="prob_3.html#T4" target="_self">Th4</a></i>
;<br/>


</div>
