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

<span class="kw">let </span><font color="Maroon" title="c1">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="finance1.html#K6" title="FINANCE1:func.6">set_of_random_variables_on</a> (<font color="Olive" title="b1">F</font>,<a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>)</span>)</span><br/>  for <font color="Olive" title="b3">phi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Olive" title="b3">phi</font>,<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>,<font color="Olive" title="b4">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Olive" title="b1">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="finance1.html#K6" title="FINANCE1:func.6">set_of_random_variables_on</a> (<font color="Maroon" title="c2">F</font>,<a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>)</span>)</span><br/>  for <font color="Olive" title="b2">phi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Olive" title="b2">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="finance1.html#K6" title="FINANCE1:func.6">set_of_random_variables_on</a> (<font color="Maroon" title="c2">F</font>,<a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">phi</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Olive" title="b1">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Olive" title="b2">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">phi</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Olive" title="b1">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/>









<a NAME="E1:46"/>
 <a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#T28" target="_self" onmouseover="rs('finance2/T28')" onmouseout="rh()">T1</a></span>;<br/>
<a NAME="E2:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span> is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#T26" target="_self" onmouseover="rs('finance2/T26')" onmouseout="rh()">Th8</a></span>;<br/>
<a NAME="E3:46"/>
 for <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font></span><br/>

<a NAME="E1:46_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:46_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">phi</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="finance1.html#K8" title="FINANCE1:func.8">ElementsOfPortfolioValueProb_fut</a> (<font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">n</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D5" target="_self" onmouseover="rs('finance2/D5')" onmouseout="rh()">Def5000</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:46"/>
 <a href="finance2.html#K8" title="FINANCE2:func.8">RVElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c4">phi</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c5">n</font>) is   <a href="random_3.html#NM1" title="RANDOM_3:NM.1">random_variable</a> of <font color="Maroon" title="c2">F</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
