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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1;<br/>
<a NAME="E1:48_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>;<br/>
<a NAME="E2:48_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#D10" onmouseover="rs('finance1/D10')" onmouseout="rh()">FINANCE1:def 10</a></span>;<br/>
<a NAME="E3:48_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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>;<br/>
<a NAME="E4:48_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D6" target="_self" onmouseover="rs('finance2/D6')" onmouseout="rh()">Def5001</a></span>;<br/>
<a NAME="E5:48_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a></span>;<br/>
<a NAME="E6:48_1_1"/><span class="lab"><font color="Green" title="E23">J1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1_1_1"/><span class="lab"><font color="Green" title="E24">B0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:48_1_1_1"/><span class="lab"><font color="Green" title="E25">B2</font></span>: 
<span class="p1">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D6" target="_self" onmouseover="rs('finance2/D6')" onmouseout="rh()">Def5001</a></span>;<br/>
<a NAME="E2:48_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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>;<br/>
<span class="kw">hence </span><a NAME="E3:48_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#D10" onmouseover="rs('finance1/D10')" onmouseout="rh()">FINANCE1:def 10</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>
<a NAME="E3:48_1_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p3">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance1.html#K10" title="FINANCE1:func.10">ElementsOfPortfolioValue_fut</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c6">G</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</a>, <a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E24">B0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1"><span class="lab"><font color="Green" title="E25">B2</font></span></a>, <a class="ref" href="series_1.html#D1" onmouseover="rs('series_1/D1')" onmouseout="rh()">SERIES_1:def 1</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>
<a NAME="E7:48_1_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:48_1_1"><span class="lab"><font color="Green" title="E22">J0</font></span></a>, <a class="txt" href="#E6:48_1_1"><span class="lab"><font color="Green" title="E23">J1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:48_1_1"/>
(  <a href="finance1.html#K11" title="FINANCE1:func.11">PortfolioValueFutExt</a> (<font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>) is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> &amp; (  for <font color="Olive">b<sub>1</sub></font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finance1.html#K11" title="FINANCE1:func.11">PortfolioValueFutExt</a> (<font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>) iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Partial_Sums</a> <span class="p2">(<span class="default"><a href="finance2.html#K9" title="FINANCE2:func.9">RVPortfolioValueFutExt_El</a> (<font color="Maroon" title="c2">phi</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c7">w</font>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c1">d</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"> verum</span><br/>


</div>
